## Real Analysis

### Guest Book

The Interactive Real Analysis (IRA) Guest Book shows comments that
readers have left (email addresses were removed to protect privacy).
We encourage you to leave your own comments, suggestions,
or corrections. To do so, please click on the

`Comment`icon and fill out the form that appears.- 1895. Jason Schorn
- I would like to commend you the exceptional quality of this website, I have spent alot of time reading through your site while taking my first analysis course. Thanks Jason Schorn

- 1894. Adrian Howard
- Just found it and am reading it through.

- 1893. min-il park
- This site is really real analysis' site!

- 1892. Carlos Bageta
- He tenido la oportunidad de buscar informacion y he quedado satisfecho con esta propuesta interactiva. Sigan adelante y mucha suerte.

- 1891. Brian Glover
- This was quite a find. What I enjoy most are the Java Tools. Thank you for putting this together and for putting on the web. Sincerely, Brian Glover

- 1890. S. Rajendra
- I have not yet evaluated yet.

- 1889. Haider Al Baghdadi
- I've been looking for something like this for almost a year. So I'm glad I have found your site now. I will be a very frequent visitor.

- 1888. Vincent Vecchio
- very nice

- 1887. sam mccartin
- The proof you give for the infinitude of primes is incorrect

- 1886. Muhammad Nawaz Naeem
- 1885. Kyle Butler-Moore
- On irrationality proof of Pi: When the equation for G(x) is meant to be defined, the equation for G''(x) has been inserted. Also, it is not clear in the presentation (although I finally 'got it') that Pi*(G(1) + G(0) ) is an integer.

- 1884. S. H. Cullinane
- Found the layout of your site of great interest, as I have a math site myself, though in combinatorics rather than analysis. (see http://m759.freeservers.com/)

- 1883. gavin bright
- 1882. Vickie
- 1881. alon
- Why the book doesn't talk about nonmeasurable set.

- 1880. BADRAOUI Salah
- 1879. shaopeng
- 1878. kim jaeho
- no comments~

- 1877. Butchie
- If you watch a blue cloud on a gray day it does look almost red. Always remeber the frog only leaps it's highest when being chased. Have a intresting day- Albert Einstein dootcher@hotmail.com rules!

- 1876. Rafael Gatica Sanchez
- 1875. Jason Schorn
- I would like to commend you the exceptional quality of this website, I have spent alot of time reading through your site while taking my first analysis course. Thanks Jason Schorn

- 1874. Johan Koopman
- Great site! Can't wait to have it on paper.

- 1873. Luis Garcia
- 1872. Greg Bernstein
- Looks good!

- 1871. Gordon Kenyon
- I have wrestled with set theory and proofs for years. I am a professional data analyst, I've taken all of the basic college-level single-variable calculus and most of a semester of graduate-level probability theory, but some of the abstract fundamentals remain out of reach. This site seems to be a step toward allowing bright-but-obtuse souls the opportunity to shore up certain fundamentals. Certainly oral instruction in these topics has failed vastly more potential statisticians than it has helped.

- 1870. Suresh Chandrasekaran
- Thank You

- 1869. Alex Yaeggy
- 1868. Johan
- Great site, although I would love to such a clear course in a book (pdf) form. Cheers, Johan

- 1867. hakan
- 1866. Debra Otto
- I wish I knew about this site before!!

- 1865. Gordon Moore
- Like the applet - really helpful

- 1864. FERNANDO FOUZ
- 1863. V.B.Lakshminarasimhan
- This is an excellent site. Being a learner of RA, I find it very useful. I never new a proof existed for Mathematical Induction itself!!!! vow!!!!! I wish there is some facility where you can accept our problems or doubts and then answer us thru e-mail and add it in your question bank topicwise. Hats off to you for this wonderful presentation !!!! Yours sincerely V.B.Lakshminarasimhan

- 1862. Mahmoud Elhaddad
- 1861. Luis Garcia
- 1860. Thomas D Browning
- I want to do this for lower leverl math and more on a elementary instruction aproach. I have been looking at some software for accomplising my goal. However, nothing I have seen comes close to what you have been able to do here. Did you discover that JAVA was the only way to effectively accomplish this level of interactivity? I am no a JAVA programer. Just entry level HTML. Do I have to lean to do my own programing in JAVA. I have been looking at: MATHTYPE, which is an equation editor. Also, I looked at NetEquaPublisher and editor. But I want the level that you have on your site. Any feed back at all will be appreciated

- 1859. Geoffrey Dillon
- This is an excellent job and a very worthwhile effort! Kudos to the author(s) and anyone else involved!

- 1858. Harold L. Clayton
- Your presentation of analysis through the internet is refreshing. Having studied mathematics at the State University of New York@Buffalo in the 1970's I find it gracious of your institution to put together such a well thought out educational package!!

- 1857. gilberto brandao
- Dear IRA This is one of the best pages of Mathematics on the Internet. I am very thankful for your great job . My interests are real analysis, topology and number theory, but only as an amateur, since I am just an AutoCad drafter with great passion for Mathematics, though I am not a Ramanujan. I wish we had more math pages like yours on the net, but unfortunately, I just found out a couple of useful pages among millions. Keep up your excellent work, I am sure that are more guys that browse through your pages but for some reasons don't sign you guest book. Well, I've been studying your pages for years, and just now I decided to sign the guest list. I really appreciate your web site. Take care Gilberto Brandao 1504 High Crest Ct Irving, TX 75061

- 1856. Robert Palmier
- WEL DONE

- 1855. Matt Burbine
- I noticed a small error in the very first Exercise set 1.1.2. The first question asks what certain sets are (in words). The first two in question are given in the answer section as even and odd integers, but the sets are defined as 2n and 2n-1 where n is an element of the natural numbers. How do you get the negative integers?

- 1854. Neelam
- I'm just beginning....... But looks like a great idea

- 1853. yvonne magsayo
- 1852. Jim Hendrick
- I think this is a great resource, well done. In my browser none of the gif's from "../symbols/" (e.g. ../symbols/{a_j} _j.gif ) are being displayed. Is there someting I can do to correct this?

- 1851. Juan Vidal
- 1850. Deborah
- This site is AWESOME! I am nearing the end of my first term of Real Analysis, and I think this site will help me a lot in my next term!!! I like it a lot, do you have any way to find key words on the site? DA

- 1849. Jon Collis
- Well thought out site. I have used it several times to help me with my hw. The proofs are nice are quite helpful. Also, the meta tags help me to find this from the search engines. Also, all the pages are linked nicely.

- 1848. wahab
- 1847. Colin Kay
- 1846. Nick Hann
- This is great! I've been hoping that US universities would be more open in publishing advanced mathematics material on the internet. I look forward to using the resource your are providing in the future.

- 1845. Douglas Boyd
- 1844. Douglas Boyd
- 1843. Puneet
- 1842. poornasree
- 1841. Balzano
- 1840. Terri
- 1839. Arlene
- This was a great site to use for my research paper on Johan Bernoulli. Thank you..

- 1838. Balzano
- 1837. basak
- ý would be pleased if ý could find metric spaces..please give me information about this subject.

- 1836. fathei
- 1835. Anthony Quinn
- Your initial message gives the impression that the download is free. Now I am being told that want $17.00. Either the download is free or or it is not free. Which is it?

- 1834. Joe Faust
- I'll be thoroughly exploring your site and will comment again later. Thanks.

- 1833. Doug Doucette
- 1832. newman
- this book is really well explained and fabioulusly done , i personaly learned a lot from it and would love to keep it as a future reference ..

- 1831. Doug Doucette
- 1830. A.White
- 1829. Roberto Fecchio
- 1828. Kristine Beese
- this is great thanks..

- 1827. A.White
- 1826. Yan Rudikh
- 1825. Celeste Carroll
- 1824. Yan Rudikh
- 1823. Alfredo Estrade
- You have a guest from Uruguay!!!! Aguante Uruguay nomas!! Campeones en el 30 , el 50 y en Core-Japon!!!!

- 1822. Leon Levy
- 1821. Bruce A. Johnson
- 1820. Magda Garza
- 1819. Larry Piet
- Greetings, Although I am not a professional mathematician, I do have a serious ambition to learn more about higher mathematics. The two years of calculus I completed as a college student cannot really serve my growing interests in science and the world in general. The problem is a lack of suitable material. Most books on higher mathematics are either far too rigorous and technical to of any value as an efficient learning tool or else they embody a popular and highly diluted approach. Your web site, Interactive Real Analysis, is exactly the kind of material that I seek. It is comprehensive without being tedious and it always gently, yet rigorously, paves the way to succeeding topics. The graphical portions are a wonderful aid in developing a visual sense of the subject. Already, I have benefited much. Your efforts at bringing this material to the general public should be commended. I only wish that more organizations could be as inventive. Sincerely, Larry Piet Detroit, MI USA

- 1818. David Hendrix
- 1817. Michael Tanigawa
- How much we engineers accept without proof!

- 1816. ranjith babu
- 1815. rodrigo fuentes
- I'm just grateful for this site. It cleared in a simple, but formal way some concepts regarding sets.

- 1814. Sergio Motejunas Ramos
- Great way to learn and free what makes it usefull for everyone. That begin of some greater.

- 1813. sathil
- 1812. Prof.O.P. Sharma
- 1811. de cleen
- 1810. Massimo Piattelli-Palmarini
- Great site. Congratulations

- 1809. Thomas D Browning
- My first visit

- 1808. A. SHABANI
- 1807. Emil Moldoveanu
- 1806. A.SHABANI
- I WANT TO UNDERSTAND

- 1805. emil
- 1804. maureen hoch
- 1803. Leah
- I would just like to say that this site has been immensely helpful in aiding my understanding of Mathematical Analysis--no other website is even comparable!

- 1802. Katrina Staley
- 1801. zhangkunlong
- a good book

- 1800. chandra
- wish i had such tools whe i studied this!

- 1799. Ben Torbenson
- 1798. Richard Cangelosi
- 1797. anne marie
- what a useful site. the examples are extremely helpful

- 1796. Hsiao Ya-Hui
- 1795. George Cole
- 1794. Donna Webb
- 1793. v.v.l.n.rao
- I AM A MATHEMATICS LECTURER FOR PRE-UNIVERSITY. I LIKE MORE INFORMATION ABOUT REAL ANALYSIS AND RECENT DEVELOPMENTS.

- 1792. burakozyilmaz
- 1791. mark anmar banlawe
- i'll try your site

- 1790. mark anmar banlawe
- i'll try your site

- 1789. v.v.l.n.rao
- I AM A MATHEMATICS LECTURER FOR PRE-UNIVERSITY. I LIKE MORE INFORMATION ABOUT REAL ANALYSIS AND RECENT DEVELOPMENTS.

- 1788. hristina
- very helpful site for analysis students!

- 1787. FELICIA MARIE HEGEDIO
- 1786. mendonca, maria de gracia
- 1785. Tracy McBride Barr
- 1784. Kimberlee Kenney
- 1783. Juan Carreño O
- Hello.

- 1782. Te Yang
- 1781. neuza
- 1780. Samir Kagadkar
- I thoroughly enjoyed your website. I loved the Java utilities etc. Samir Kagadkar ( B.Tech Electrical Engg.) IIT Bombay, Bombay, India

- 1779. Tracy McBride Barr
- 1778. Anne Marie Yusko
- I've just stumbled upon your site. Can't wait to see all the "juicy" details!!

- 1777. valentijn
- 1776. Shona Schwab
- Thank You!!!!

- 1775. Jonas Persson
- A good comprehensive complement to traditional (printed)book on the subject. I've used it as a complement while taking a course in Measure Theory at Uppsala University, Sweden.

- 1774. rajender
- 1773. David Nementzik
- 1772. Jeff Johnson
- 1771. Curtis Smith
- 1770. Andrew McLintock
- 1769. E Davis
- 1768. Vicki Adams
- I find your site to be extremely helpful in explaining many concepts my Advanced Calc. book does not go into in depth.

- 1767. Aissa Aibèche
- 1766. Mario Cavani
- 1765. TheCag
- I was looking for information one Euler's Number. I am trying to get some insight into this number sequence. Some of what you have on this website is helping me better understand this subject. Thanks.

- 1764. Brad
- 1763. Massimo Pellegrino
- 1762. Brad
- 1761. daniel
- 1760. Dwayne Fuller
- My, my, my... this is indeed a technological tool. Who thought that you could put math on the web.

- 1759. shenhong
- 1758. ponce
- 1757. Michael Hendricks
- 1756. Paul Wang
- cool site my stupid teacher will be impress now!!! and suprise

- 1755. Celeste Carroll
- 1754. Ambika
- Great website, I happened on this, and it's really making my taking analysis this quarter a lot easier, the proofs are easy to follow, very straightforward

- 1753. Russell Simmons
- 1752. Ariel Luzzatto Ph.D.
- very friendly, clear and useful

- 1751. Ana Isabel
- A very interesting material.

- 1750. william lanzo
- 1749. Dusan Bednarik
- This is a good idea.

- 1748. Rui Veiga
- 1747. Francisco Escareno
- 1746. Matt Musson
- I'm in my third year of a four year BA in honors math at the University of Western Ontario. I've had your site bookmarked for a couple of years now. Everytime I encounter soemthing I failed to understand the first time around, I consult the site and always gain a better understanding of the material. The site is well presented and extrodinarily informative.

- 1745. joão machás
- As a first year student at the University I have been learned a lot with your clear explanation of Interactive real. Thanks a lot João

- 1744. sofia figueiredo
- 1743. rodney wong
- 1742. nachimuthu.n
- your web is very very really great.

- 1741. vasco rodrigues
- I really like your site, it as helped me a lot. Thank you very much and keep on with the good work. Kind regards, Vasco

- 1740. Trudy Tyler
- 1739. khan
- thank you very much for this online text... it really helps to understand these subjects...

- 1738. Tom Reuterdahl
- 1737. M M
- 1736. saeed
- 1735. Valeria Pardini
- 1734. Ravindra Kumar Puli
- Really nice to see such a course innovative and basics are well covered

- 1733. khan
- thank you very much for this online text... it really helps to understand these subjects...

- 1732. Tom Reuterdahl
- 1731. Sung-youn Kim
- 1730. MICHAEL R. PACAYRA
- 1729. Catherine Sena
- Do you have an online tutor? I really need help in my real analysis class.

- 1728. Bhatt Rajanikant
- 1727. ehsan
- 1726. leonhard euler
- 1725. Tom Thurman
- This work is much appreciated!

- 1724. Jim Trejo
- I am currently taking a required mathematical analysis course, and I could use some help/pratice. Thanks.

- 1723. Sung-youn Kim
- 1722. Joao Pinto
- 1721. David Orr
- 1720. bill Capps
- 1719. Tony Martin
- 1718. Jack Crowley
- 1717. Shrikrishna M. Badiger
- It will be very nice if I can get whole book in pdf format.

- 1716. Jayachandran Kamaraj
- hi Iam an engineering student. I was first worried when i attended the probability class and found that it was all abstract with the other math doctorate students. But this website helped me a lot by the defining the fundamentals which i needed very badly. Thank you very much

- 1715. Edward Moyle
- This is a good idea.

- 1714. patrick reardon
- 1713. Will Browne
- Really great reference for when my text doesn't have everything that I want to know. It helps me as a quick way to look up definitions and other things scattered sparsely through my text.

- 1712. Lijian Chen
- I am a student now in a real analysis course. I want to improve me level over using your website. yours sincerely lijian chen

- 1711. Peter Joseph
- 1710. Feri Farassat
- 1709. Leanna House
- This is great! I hope that you can finish chapters 8 - 10 before my course ends!

- 1708. David D. King
- 1707. Ray Rivera
- 1706. trudy makhaya
- Wow, this is brilliant. I am econ graduate student and I have just taken a course that presumes knowledge of real analysis. Having not done any RA, I have been quite stressed. This is a good, fun way of learning the subject. Thank you for the service.

- 1705. Amanda Wachsmuth
- This online textbook has been and continues to be a great reference and review basis as I grapple with graduate level Modern Analysis. It's also nice to know that another Wachsmuth has forged the path ahead of me.

- 1704. Miguel Sánchez
- 1703. Peter McPolin
- 1702. andre duchene
- this is a fantastic addition to my textbook. Thanks

- 1701. Simon See
- 1700. fei
- your web site is so helpful. i really appreciate your service

- 1699. Camelia Kuhnen
- 1698. kevin washington
- 1697. methuselah k. arenas
- hello peps behind this site, i thank you for this wonderful site i t really helps me in my research works which rquires to do some internet sources thanks you so much and keep up the good work!

- 1696. uzma reem
- 1695. dan yotz
- I dug the historitcal section, it is somewhat comforting to realize that I am not the only one who has been driven insane in the pursuit of infinity. Alas, the basis of my work seeks to prove that negative infinity does not exist. Thereby overturning Calculas as we know it. Of course we already know that zero is completly theoretical and does not exist. Proof of such leads to the proof of negative infinity. Time continues, existance of either would require the possibility, with a high probability that everthing is motionless...again completely theroretical. This also leads to the proof that the earth is actually at the center of the solar system. A geocentric model also requires the stars to be the Sun. Heliocentricy does not allow for the infinte, as infinity over infinity would always equal one.

- 1694. Michael A Blischke
- A good reference site. I've bookmarked it.

- 1693. Barbara Benny
- 1692. Angela Betts
- 1691. Paddy Ewer
- Let me look at it first

- 1690. Thabani nyathi
- 1689. Kalina
- I really like this page. How much time did it take you to finish it?

- 1688. Abdenour Achab
- THANK YOU

- 1687. Gerry Moultine
- 1686. william lanzo
- 1685. Emily
- For an even integer n and n>2 prove ((2 to the n power) -1) is not prime.

- 1684. sivakumar
- The useful service which give lot of information thanks

- 1683. Fei Chun MA
- Thanks very much for putting such a great material to the internet. I really benefit from it a lot! However, it will be perfect if Chapter 8 and 9 can also be browsed online.

- 1682. Byron Shoop
- Thanks. A shining example of what the web is supposed to be about.

- 1681. sue brink
- 1680. Dr. István Ratkó
- This is very good.

- 1679. leonid
- 1678. Joe
- 1677. neslihan celik
- 1676. Keith Mc Clung
- thanks

- 1675. yoonmee ham
- 1674. Mick Horn
- 1673. michael ahming
- 1672. sriram
- hi there! Wonderful job. Though I must say i am quite ignorant of the stuff. I hope to learn somethings from your valuable tool. thanks sriram

- 1671. vitaliy
- I find the web site very useful for studying real analysis

- 1670. Felipe Borges Cunha
- Hi, everybody!!! I do not know what Real Analysis is but I think I am about to discover... Thanks, Felipe

- 1669. Marilyn Brown
- 1668. Jodie Sovak
- 1667. Vikas Yadav
- Hi, this is a fabulous site. I am doing a course on Advanced Caculus, which is basically about analysis. And, being an electrical engineer (and not a mathematics major) I have found this site extremely helpful in getting basics correct. Thanx a lot for this great effort. Regards, Vikas Yadav

- 1666. Hoang Nguyen
- 1665. Jay Carbonella
- Great set up. Very helpful and easy to navagate. Jay Carbonella Student University of Maryland

- 1664. Wayne Gabledon
- 1663. Christopher Valle
- 1662. Trudy
- I was looking for some proofs on math induction and proving the rationality or irrationaliyu of certain numbers

- 1661. victoria mata
- 1660. Frank Giarratani
- 1659. Joachim Stoeckler
- Great project! If you don't mind I will tell my students (4 semesters of calculus for physics and engineering students) to visit your page as a companion to my course and to come forward with questions relating to your material. I will not copy any material from your website, as you request in your copyright notice. Please let me know if you are aware of a similar project in Germany. Best regards, Joachim Stoeckler University of Dortmund, Germany

- 1658. agnes
- 1657. Frank Scardetta
- Very refreshing to see this site. It has made the difficult corse so much easier to understand. Thanks again.

- 1656. Dan Phelps
- Good complement to rigorous proofs.

- 1655. Joachim Stoeckler
- Great project! If you don't mind I will tell my students (4 semesters of calculus for physics and engineering students) to visit your page as a companion to my course and to come forward with questions relating to your material. I will not copy any material from your website, as you request in your copyright notice. Please let me know if you are aware of a similar project in Germany. Best regards, Joachim Stoeckler University of Dortmund, Germany

- 1654. agnes
- 1653. Dan Phelps
- Good complement to rigorous proofs.

- 1652. tarcisio praciano-pereira
- 1651. Jorge Ramirez Arroyo
- It was good!

- 1650. Jerry Easley
- 1649. fezzardi marino
- very good thank you!

- 1648. Lawrence Millen Madeira
- 1647. Joel Covarrubias
- Just wanted to thank you for this wonderful site. I'm a programmer professionally and a mathematician amateurishly (?), and I stumbled onto your site almost accidently. The coverage of topics is thorough, the glossary is fairly complete, and the organization of your site is very clear and intuitive. Thank you again for this well-designed resource. Joel C.

- 1646. LALITHA .V
- GOOD WORK!!!

- 1645. alma gruver
- 1644. KO'SHON
- 1643. nicco
- 1642. percell ridges
- 1641. percell ridges
- 1640. nicco
- 1639. Gabor Lukacs
- I am a PhD student at York University, Toronto. This semster I was assigned to lecture a first year course in calculus for math major students. While preparing my notes for the lectures I realized that I do not know how "limsup" is called in English (I am Hungarian, but studied all the math in Hebrew). So I went to the web, and found this wonderful site. I am seriously considering referring my students to this site, because it covers more-or-less all the topics that I am going to teach. I think that the site is very well-constructed - I wish it were mine. Its author can be very proud of it. Best regards, Gabor Lukacs

- 1638. TRUTNEV
- 1637. Young hong
- Hi Whe are you going to complete your extremely helpful site?

- 1636. M. Patricia Ortiz
- 1635. TRUTNEV
- 1634. Young hong
- Hi Whe are you going to complete your extremely helpful site?

- 1633. Gabor Lukacs
- I am a PhD student at York University, Toronto. This semster I was assigned to lecture a first year course in calculus for math major students. While preparing my notes for the lectures I realized that I do not know how "limsup" is called in English (I am Hungarian, but studied all the math in Hebrew). So I went to the web, and found this wonderful site. I am seriously considering referring my students to this site, because it covers more-or-less all the topics that I am going to teach. I think that the site is very well-constructed - I wish it were mine. Its author can be very proud of it. Best regards, Gabor Lukacs

- 1632. M. Patricia Ortiz
- 1631. Robert Birchfield
- 1630. ISAAC CHEPKWONY
- 1629. Clark Alexander
- This is a well done sight. It's good for me just to brush up on some topics in Real.

- 1628. David Thompson
- 1627. Kord Eickmeyer
- A very interesting and helpful collection! Thanks for providing this service!

- 1626. Vu Hoan Pham
- 1625. Irving Fenichel
- 1624. Hari Pulapaka
- Really well done. Analysis is not my specialty, but I hope to use this site as a reference. Thanks for doing this. - Hari Mathematics and Computer Science Stetson University DeLand, FL 32720

- 1623. m early
- Please e-mail ordering details of the e-book. Is a hard copy version available?

- 1622. Joseph G. Harrison
- 1621. hsiung
- 1620. potnuru kiran
- 1619. leonid
- 1618. sclee
- 1617. Keith Birkinshaw
- 1616. don costello
- 1615. ngoc le
- 1614. Richard G. Allen
- 1613. jeff
- Please, please do a construction of the reals...

- 1612. Al Mannon
- One of the best analysis sites I have found. Easy to navigate and the presentation is easy to follow. Thank you for your excellent work.

- 1611. deb bledsoe
- this is a cool way to learn something that I have no earthly reason to know, but think is interesting anyway.... and it sure beats watching some dumb tv show, although it's not quite up to being in a cave on a Friday night... but if I have to be here, this is great (even though I'm not a math person, I am logical and I actually find myself understanding everything but the notation.... and I'm even starting to get that ;) ) whoever is responsible for doing this, keep up the great work... the trailer lady

- 1610. leonid
- 1609. sclee
- 1608. Keith Birkinshaw
- 1607. Paul Schweinzer
- Really well done--congrats!

- 1606. Francislei Caxeixa
- 1605. Jev
- Great initiative! I'm a student in physics at the Delft University in the Netherlands, and the book we are using for the differential equations really sucks so much that i was forced to look on the web. This is the best i've found so far.

- 1604. Diana Mecum
- This seems to be a very exciting online site, and I look forward to exploring it further. Thanks for the effort. Diana

- 1603. vnkrishna
- I would be nice if other topics like probablity theory,random variables, random process are included.

- 1602. Agustín Pacheco Cardenas
- 1601. ana v. moreno
- Prometo enviar mis comentarios en cuanto lea algo de este texto. Gracias por ponerlo a mi disposicion! Vicky.

- 1600. montaudo
- I find it a nice project

- 1599. Doug Jones
- I have just found your site, and I am looking forward to reading it.

- 1598. Tom Walsh
- Theorem 1.4.4 defines the rationals by saying "Let A be the set N x N-{0}..." It seems like you mean "Let A be the set Z x Z-{0}" First of all, the naturals do not include zero (so the restriction on the second term isn't needed), and second of all, this would only serve to define the positive rationals.

- 1597. jose
- very useful

- 1596. Mark Scheffer
- Good source of information, the material is pretty detailed and concise. Keep up with the good work!

- 1595. John Hatherley
- 1594. Trevor Newhook
- I just found this site but I'll be looking it over to prepare for my Real Analysis course in the fall.

- 1593. Terri Grant
- 1592. Yu Zhang
- It's really a wonderful place for math study

- 1591. Victor Kowalenko
- 1590. Mauro Donizeti Tonasse
- Cogratulations. It is a nice site! M. D. Tonasse

- 1589. Manfred Zimmermann
- 1588. Geoff Davis
- 1587. JOSE LUIS COLORADO H.
- Really Amazing! This is my first time looking into advanced calculus and analysis, so I am looking for clear introductory material. After to read Spivak´s book for some topic, I see in your tutorial thus the concept is very clear. congratulations you are very good web-writer! jose luis colorado

- 1586. John Ryan
- When is the download ready?

- 1585. a.rhman bashir
- compytional numercal anlyses

- 1584. Hélio Carneiro Ferreira
- 1583. Ana Isabel de Azevedo Spinola
- 1582. Professor Gray
- Is it possible to download this text. This is a beautiful text.

- 1581. Bruce Graham
- 1580. Bill
- Thank you for this free site!

- 1579. Jeff Webb
- Interesting site, I will mention it to my students. Note for your development work: |f| Lebesgue-integrable and f measurable imply f is L-integrable. Theorem 3.3.4 (B-W thm) in the proof one should choose j_1<j_2<j_3...

- 1578. Uhshbah Pervez
- 1577. Anne Winkelman
- 1576. John Matthew
- 1575. Terry Sontag
- I am a senior undergrad in mathematics at Michigan State University and I am REALLY glad that someone has put a real analysis resource online!! Thanks to all who had a part in puting this online!!

- 1574. Manfred Becker
- 1573. Aracelis Geraldino
- What marvelous discovering, it's fabulous help. Thank you so much.

- 1572. Francesco
- I find your work highly enjoyable. I just discovered it and, as first thing, read the chapter on integration and found it clear and stimulating: surely I'm encouraged to read the other parts of the book. Thanks for having let your work avalaible on the net, it's a great idea. Francesco, Physics student in Rome

- 1571. panda , S.N
- Let me first see . regards

- 1570. Rabi De
- 1569. Carlos M Kier
- 1568. Marc Forget
- 1567. dgreenig
- 1566. Ali Ulger
- 1565. Catharine Lister
- I am doing a Mathematics degree with the Open University in Yorkshire.Will have a closer look at your site when I have a little more time,it looks interesting.Hope it will help!

- 1564. morne holland
- 1563. Frederick Akalin
- 1562. Josh Isralowitz
- Very nice work I must say. A should also state that I am an avid fan of the Seton Hall radio station 89.5 (for some time now, maybe 6 years.) I must say also that if it would be possible to actually post up audio as a lecture to aid each portion of the text, that would be an incredible tool for someone interested in self study. Make a charge for something as this maybe $30 or so, and maybe make a small profit for it. I am sure many people would pay for such a thing (as I assuredly would.) Perhaps I will email the maker of this site my comment too. Nevertheless, this is a wonderful site you run thank you JOSH

- 1561. GUOXU
- vERY GOOD SITE!!

- 1560. David Revell
- 1559. Jesus Oliver
- I have given a look to this e-book and it looks great. I think this kind of presentations are ideal for those who want to review some forgotten concepts. Thank you very much

- 1558. Susan Randall
- 1557. Thomas D Browning
- Just got here will comment later

- 1556. chan
- I am using this to learn the fundamentals of Real analysis.

- 1555. amir
- 1554. Felix
- 1553. R. F. McKenzie
- Interested in Green's Theorem

- 1552. richard fell
- 1551. Josipa Milisic
- 1550. joop van broekhoven
- I should have had this nice course back in 1962 when starting with my physical chemistry study. I was quite fond of mathematics at that time especially analysis and vector analyis. The only part that I am still using regularly for may work is numerical integration of differential equations (I develop industrial chemical processes). I passed your page to may youngest son who is sorry that he does not need this beautiful stuf for his current study of accountancy regards, Joop van Broekhoven

- 1549. mexas s
- 1548. David Favis-Mortlock
- Very nicely done! But note that in my part of the world (Belfast, Northern Ireland) the acronym IRA has a rather different meaning, in general...

- 1547. Manuel Rosa
- 1546. Frans Kerstholt
- I am just starting my first visit: so, no comments FK

- 1545. Marta garcia-Huidobro
- 1544. ERICK ROBLES
- EXCELLENT!!!!!!!!!!!!!

- 1543. ajit kumar
- This is very useful online set on nets on real analysis.

- 1542. Ray Tenebruso
- Thank you for all of your work on this project.

- 1541. lawrence bird
- 1540. DaJun Kang
- 1539. chenyj73
- Thank You!

- 1538. Sokol
- 1537. mohamed
- real

- 1536. Reeta Shukla
- 1535. Maksim
- 1534. DR.LOVNEESH GARG
- just stumbled upon your beautiful site while searching for material on Zeno's paradox; looks interesting and informative; will like to go through it for some time before opining more about it.

- 1533. Douglas Lee
- 1532. rodrigo flores
- 1531. Fern
- Your site is making studying for my Honors Calc final much easier. I especially like all the examples you have worked out. Thanks!

- 1530. Duke
- Nice!

- 1529. GEORGE ANTASHYAN
- DEAR ADMINISTRATION, I AM INTERESTED IN APPLYING IN TO A GRADUATE(Ph.D) DEGREE IN MATHEMATICS. ONE OF THE PREREQUESITES IS ADVANCED CALCULUS. PLEASE HELP ME FIND A SHORTEST WAY TO COMPLETE THIS COURS. THANK YOU. SINCERELY, ANTASHYAN.

- 1528. Manjunath
- Its nice to know that there are people who are generous and kind enough to put up Analysis in a simple fashion after taking lots of trouble... that which is simple is alos beautiful!

- 1527. Mat
- This is an absolutely incredible publication. I wish I had this available in my first year maths class!

- 1526. Matt Boelkins
- a nice surprise that turned up during an otherwise unfruitful web search. thanks for your efforts. Matt Boelkins Dept. of Mathematics Grand Valley State University Allendale, MI 49401 616 895 3384

- 1525. Roseann White
- 1524. Danilo De Rossi
- Hi, well done, I appreciate very much the effectiveness of your site. I'm an italian student, and I've realized from few months that to learn calculus I have to look thru sites like this. Best Regards Danilo

- 1523. Kim JaeMyung
- Very good site. I have got a much help in this site. Thanks.

- 1522. Nafeez
- I cannot begin to thank you enough for this wonderful site. I needed to review some Real Analysis for graduate work and your site is better then any text imaginable. I only wish that this site was available when I was taking Analysis in undergrad - Oh what a nightmare. Once again Many thanks Nafeez

- 1521. jaime joya
- ¡excelent site!

- 1520. francisca
- 1519. Lori
- Thanks so much, this website has been a huge help.

- 1518. Eunkyeong Lee
- ...

- 1517. Thomas Russano
- 1516. Elsa Mendieta
- 1515. kevin orr
- 1514. Jacob Mampane
- 1513. sumantamukherjee
- I had visited this ealier the section on topology has been useful for me at my curriculum at IIT delhi

- 1512. nicholas buck
- just looking; i shall return

- 1511. GUSTAVO SANCHEZ
- I'M AN ELECTRONIC ENGINEERING AND THIS PROJECT REALLY HELP ME TO UNDERSTAND THE MESAURE THEORY AND METRIC SPACES.

- 1510. Sita Koirala
- I am going to take Real Analysis first part in fall semester. I am hoping this site will help me to get prepared ahead. Thanks for your hard work and good information.

- 1509. Sergio Plaza
- 1508. chris dods
- this is a wonderful resource, to which I frequently refer. keep up the good work.

- 1507. Paul S. Lazar
- This is great. I love it !!!

- 1506. Alojunya
- Thank you for providing a good resource for math students. Keep up the good work. Much appreciated.

- 1505. Robert James
- 1504. horizonte
- no commnents for while.

- 1503. Linu M Koshy
- 1502. Osman
- 1501. frank fritzsche
- very nice page

- 1500. Carlos A. Scarabino
- I didn't see the course yet. I'll send you my comments later.

- 1499. Richard Fratini
- Excellent!!! Keep up the good work! Congratulations.

- 1498. Horacio Arango Marin
- Felicitaciones.El contenido es exclente. Deberian agregarse mas graficas y anexar un capitulo sobre distribuciones

- 1497. jean Johnny
- 1496. Plakias George
- 1495. Manish Ahuja
- cool site , Make it free if u can.

- 1494. Abigail G.
- i find this site to be of tremendous help with my real analysis. This site is just wonderful and easy to follow through and easy to understand the concepts. thank you so much for putting up such a great site.

- 1493. Jim Summerlin
- 1492. Franz Gerle
- 1491. keith Conrad
- 1490. Evan
- 1489. Adriaan Ruiter
- 1488. Yoek Nam Khor
- Wonderful tool. Complements my Real Analysis class here at Berkeley.

- 1487. Tuna Çelik
- That is the perfect treasure for any mathametician to learn the basics of mathematic. When i found this site in the search engine, i didn't know what i am found. However, i now can understand the importance of this site! I have been studying my Exams with this site... GREAT!!! GREAT!!! GREAT!!!

- 1486. Jennie
- 1485. Claudia Rosana Martínez
- quisiera saber si este curso interactivo no está disponible en español, ya que mi inglés es muy pobre. Soy profesora de matemática y Física y me interesaría mucho poder realizarlo en mi idioma

- 1484. Evandro Sena Freire
- 1483. Gustavo Solorzano
- I will take a course on Real Analysis the next year. It looks very difficult, so I'm very glad that there is such a web page. I'm sure, I will use this page houndred of times. May/2001

- 1482. rob seaman
- 1481. kibeom kim
- thank you!

- 1480. Gavin Thoms
- 1479. Frank Barker
- Delighted to meet you

- 1478. Frank D. Rogers
- Great set up! Will this become a pay site later on.

- 1477. Joe Faust
- Just starting, will comment later.

- 1476. Barry Nason
- 1475. manoocher aria
- 1474. dirk vrielynck
- 1473. BRIAN BOAN
- 1472. Hassan Azadi
- 1471. Giovanni Biondi
- 1470. Simon Yeung
- This web site is very attractive and very educational! I wish I had found this web site before taking my analysis class.

- 1469. K.Kassapis
- 1468. Dr RN Mukerjee
- 1467. Dennis Turpin
- 1466. Jim Main
- Thanks.

- 1465. william martin
- 1464. rich byrom
- 1463. E. A. Parent
- Love it so far. Get on with the missing chapters!

- 1462. Karin Weule
- 1461. spkumar
- since I am working as a mathematics teacher it will be of a great use for me to brush-up and update my knowledge .

- 1460. Luis
- 1459. Kendra Kahn
- 1458. Abhinandan Roy
- This e-book is an extremely valuable source of fundamental Real Analysis knowledge.The applets are very beautiful for explaining the theories wherever necessary.The pending chapters are requested to be completed soon. An advanced treatment of Real Analysis may be a very good supplement of this introductory book.

- 1457. Taufan
- 1456. Jay
- This is really great!

- 1455. mark readman
- fab site

- 1454. Thaddeus Evert
- How do you get into the site?

- 1453. Looser
- 1452. Bob Cunninghams
- I've just starting studying Image Algebra (using the text by Ritter and Wilson, as well as Ritter's notes from his University of Florida Web site) to support my work in digital color image processing. I needed only a clear and concise (verbal, non-symbolic) definition for supremum and infimum, and a Google Web search led me to your site. I was looking for a couple specifics, and found a treasure trove. Thanks!

- 1451. jonnyboy
- 1450. Maurizio Avellone
- Very helpful and very interesting Greetings from Sicily

- 1449. Melinda
- 1448. Kathy
- 1447. Peter K.H. Gragert
- I am very much impressed of the work that has been done! This an example of what an interactive book can be more then a 'normal' book!

- 1446. Szantovan Leontin Stelian
- 1445. Tim Boudreau
- This is by far a very useful tool for anyone taking a course in Real Analysis. It is very well presented, and the proofs are easy to follow. I wish I had known about this when I took the first half of the course last semester because it wouldhave been an essential tool for helping me better understand the material. Thanks for putting this on the net! Tim Boudreau

- 1444. jeffrey Carlson
- Thank you for putting up this site. I've been curious about how to construct the reals, etc., ...because I know "Real Analysis" goes deeper than the "Elementary Calculus" I done got taught for two years two years ago. In particular, I think I had better figure out what I'm talking about with real analysis before I go on to complex analysis or do much more algebra. I just have this strange sort of feeling about that. It appears shu.edu has severed your name. Sorry about that. We hope you have a nice day. I had something brilliant and relevant to say too, but I forgot it. Oh well.

- 1443. Kurt Richardson
- Very impressed! Excellent resource.

- 1442. Sérgio Silva
- I have suggested this site for all my students here in Brazil.

- 1441. villa
- 1440. sumantamukherjee
- 1439. HUMBERTO MEJIA
- 1438. Fei Gan
- 1437. K V Subramanian
- 1436. Stanislav
- 1435. Jon Ratzlaff
- 1434. halit orhan
- 1433. Jason
- This is a great site. Just having the statements of theorems available at a click is wonderous in and of itself; having examples demonstrating theorems is even better. Many laurels to all who put this together.

- 1432. Ron Duehr
- 1431. phil
- nice piece of work!!! make real analysis rather easy.....i love that

- 1430. Goknur Kaplan
- This site is really excellent.

- 1429. Richard A. Smith
- 1428. Dana
- Could i get answers and proofs to questions that i may have?

- 1427. Lasantha Gamage
- I have many problems in differant fields of Mathematics. I hope, this will help me

- 1426. joshua jones
- 1425. e m
- 1424. Cindy Henning
- 1423. Nadine Nieser
- Your sight has been very helpful to me in my Advanced Calculus course. I was looking for info on Differential Forms. Do you have any advice?

- 1422. Chenmc
- 1421. Foots
- Thanks for providing such a site

- 1420. rapee
- Best web page for mathematic

- 1419. Chris Williams
- An excellent source for Real Analysis, especially once I'm no longer in the class.

- 1418. TonyB
- Good work!

- 1417. Claude Demierre
- Look's like a great web math course, I'll have à deeper look now.

- 1416. fangxing
- 1415. Ahmad Sanei
- Very Impressive. I am using it a lot to remember the forgotten things. The applets are not very effective. You change parameters but it does not effect the picture (sometimes)

- 1414. Francis Dufour Save
- 1413. Amin Elemam
- Very good quick reference.

- 1412. Brett Foley
- I am taking Introductory Analysis as an independent study, this site was a great resource. Thanks, and keep up the good work.

- 1411. Alex
- 1410. caroline
- Brilliant site !!!!!!!!

- 1409. gary Kelly
- 1408. Jana
- You are really good

- 1407. Guy
- Very nice analysis resource. Interested in downloading the book.

- 1406. William J. Moore
- I am currently a student at George Mason University and studying this in Advanced Calculus. This site is very helpful with understanding the concepts. Thanks

- 1405. John Hegedus
- Thank you for making this information public and easily accessible.

- 1404. frank mcleskey
- 1403. Christina Mason
- Thank you. You've been a god-send for my Real Analysis course at the University of Buffalo. I currently need info on Metric Spaces. Your site is great. Thanks.

- 1402. Kim
- 1401. farrah
- Thanks for being informative and helpful. I like the way you have laid the website out, an the interactive theorems and proofs make it easy to use!!

- 1400. hari
- 1399. c s Somerville
- 1398. betelmina
- 1397. Gilbert Patt
- 1396. R (Chandra) Chandrasekhar
- Very interesting site. I was helping my son prove The Bernoulli inequality by induction and wondered why x >= -1 is a condition and discovered this site by surfing the web and seraching for the Brenoulli inequality. May such enlightened educational methods, that set distance at nought, flourish long and well!

- 1395. Hans Morgan
- 1394. Grover Prowell
- Dear Dr. Wachsmuth, I have attempted real analysis classes twice before at the University of Nevada Las Vegas, and will be doing so again for a third time next fall. I hope that with the aid of your website, I will have more success than previously (failing, changed to audit; then a struggle to obtain a C+, but without confidence in what I attempted). Grover Prowell

- 1393. jma
- excellent!! really impressive

- 1392. Vilar Figueiredo
- This site is very useful for my students. Congratulations.

- 1391. Frans
- Very wonderful site!

- 1390. Vasco Nuno Cruz Rodrigues
- Congratulations for the excellent job that this site offers to mathematicians all over the world. Keep on the good work.

- 1389. Surrendra Kumaran
- I got to this site from cut-the-knot.com

- 1388. karras
- 1387. Akintunde A. Funso
- 1386. Abel Luli
- Overall, it is helpful.

- 1385. Ulises Carcamo C.
- It is a great Idea to have this type of book available for anyone who is interested in studying Math. I think this idea would be developed in other fields such as Biology, Phisics and Economy.

- 1384. mravi
- 1383. RESEGO MOSEKI
- 1382. Dr. Ahmed Kaboudan
- 1381. Paulo Robel
- This site is helping me very much in my analysis course this year. So thank you very much

- 1380. Sandy R. Jackson
- 1379. vlora
- 1378. William A. Barr
- 1377. vivekananda
- I am a student of final year of graduation. interested in topics such as metric spaces,sequences and series of functions,compactness of set etc. I FOUND YOUR WEDSITE TO BE VERY HELPFUL. KEEP IF UP.

- 1376. SnowCrash
- Thanks for this wonderful site. I found some good information on Zeno's paradox here. I'm going to have to come back to check out the rest of the site.

- 1375. miguel angel ortiz Ortiz
- 1374. C Lutterodt
- 1373. Dominick DelCasale
- Thank you for an informative and helpful resource.

- 1372. Daniel Fugulin
- 1371. Elliott Middleton
- 1370. Greg
- Seems like it is a very nice site to get some intro to functional analysis. Actually I was looking for some discussion on Stieltjes integral. In Chapter 7 it states that 'we discuss Riemann integral first, then we move to Riemann-Stieltjes and then Lebesque. I was a bit disapponted when I couldn't find anything on Stieltjes integral after all. Hope in a future this material will be added. But overall very impressive. Planning to spend some time reading each chapter. Thanks!

- 1369. Megan Kerr
- 1368. ivan Adrian
- 1367. ed Castillo
- 1366. Renee Barter
- I think this is a great thing. Good work!

- 1365. Miguel de Guzman
- Dear Professor Wachsmuth, My best congratulations for your beautiful effort. My students and I are very glad to be able to visit your page and we hope that you will soon continue with sequences and series of functions and so on. A small remark. Please check this: The Cantor function is differentiable almost everywhere, but not at the points of the Cantor set, i.e. not on all [0,1]. Best regards. Miguel de Guzmán Madrid

- 1364. Baldur
- 1363. masud
- I read the stuffs. It's was very informative and helpful for me. Thanks a lot. Keep it up.

- 1362. Stefano Pogany
- This site is wonderful. I am a chemist interested in set theory. I was looking for books on the subject. This site is a great learning experience. It is almost like speaking with a teacher. Great contribution! Stefano

- 1361. Eugenio Meoño
- It´s very interesting your interactive system

- 1360. Juanita
- 1359. J F GUTIERREZ
- Today, 1st march 2001, I found this web page. Next days I will send you any comments. Best Regards.,

- 1358. vinay
- 1357. Minglei Hu
- A good idea, to put your book online. I am a software engineer in Tokyo. My major was Applied Mathematics when I was in the university. If you are a professor please tell the young students that mathematics is the most important subject they should learn in the university. Mathematics is the base of computer science, Java C++, they are trival when comparing to Real Analysis. (Although I get my butter and bread on such softwares)/ Can I have the whole book in Postscript, LaTex or PDF format as a whole. If you have, would you please send a copy to me ? Thanks again !

- 1356. chandramouli kashyap
- very good presentation of material. Ideal for newcomers to the field like me. I hope you would complete the remaining sections soon. Also a section on applications of real analysis (for eg, numerical methods) would help.

- 1355. Judit Kardos
- 1354. Toshiro Kubota
- Very nicely done!

- 1353. Chris Dods
- this is a wonderful resource. big up to everybody who made this possible.

- 1352. Isabelle Hemmings
- The book has really helped me understand some of the more fundamental aspects of real analysis. Thanks so much for doing this!

- 1351. joyce
- Thank you

- 1350. Kwang Hui
- Hi, real cool. I loved it....Cheers KH

- 1349. mars
- very good!

- 1348. Number Monkey
- What a terrific site. St. Georg Cantor would put his hand on your shoulder, nod approvingly, and recognize your scholarship and selflessness with a smile.

- 1347. Amre
- 1346. Jesse Jenkins
- Very nice work. Terse, direct. Thank you.

- 1345. exia bauer
- 1344. Joel Osborne
- Thank you, this is a wonderful site, great to use. I am a student at Embery-Riddle Aeronautical University, studying to become a Aerospace Engineer, and this will be helpful for all, I will add this to our class links, thank you once again.

- 1343. Omran_Kouba
- 1342. Hassan Azadi
- 1341. K Shelton
- 1340. Necésio José Faria Tavares
- 1339. Mike McKinley
- I'm an electrical engineering student at GA Tech who is interested in learing about Lebesgue integrals.

- 1338. rahul singh
- 1337. Christian Werner
- Hello! I wanted to thank the creators and authors of your Site! I am having an Analysis exam and was about to forfeit when I came across this Adress. In just a few hours I learned more than I did in days from these hard-to-understand books. The Java Applets are a great feature and the examples are well chosen. Also, one can get into each chapter easily, as the introduction is kept very simple. I hope the latter chapters are completed soon!

- 1336. Stephen Rice
- 1335. Christina Patterson
- I am interested in learning this subject and I plan on sitting in on a class at the college on campus in order to help me out while I am teaching AP Calculus at the high school.

- 1334. dante.capezzani
- 1333. erin clement
- Your site is great! I can't wait until it is finished.We're starting sequences of functions and I'd like to see your proofs and examples on the subject.

- 1332. s. robinson
- 1331. ajai
- You have made mathematics Soulful!!...what more could i say!!!

- 1330. Ana Isabel de Azevedo Spinola Dias
- 1329. William A. Barr
- 1328. M Panthi
- 1327. tom
- Outstanding!!!

- 1326. JTmoney
- This site helped so much it wasnt funny. I'm almost afraid to let too many people know about it. We are using a text in my class that is not very helpful for someone who is not familiar with analysis at all. Your site has allowed me to almost teach myself and get a pretty good understanding. I just wish it would not have taken so long to find it, or i would have been on top from the very beginning.

- 1325. Troy Parker
- Just a note of thanks for your web page on De Morgan. I'm a student at Central College in Iowa and I am studying logic in a mathematics course. Your page adds supplemental information that improved my understanding. It's people like you that make the web such a wonderful resource. Thank you.

- 1324. Ron Gagne
- I look forward to visiting your site. It sounds interesting

- 1323. ling wang
- it is wonderful! i never got so clear idea about the concepts in real analysis.

- 1322. kalesv
- 1321. Diana Baranec
- 1320. Richard Webb
- 1319. M. C. Valsakumar
- 1318. Timothy Choi
- It is amusing that we, those who study mathematics, have such a nice web site like IRA. Visual aids definitely help first timer to learn real analysis with less struggles.

- 1317. Cichon Mieczyslaw
- 1316. Thanos
- Your site is excellent for students and teachers alike.Long live mathematics.

- 1315. tim brown
- this page is a VERY informative collection of information. i have taken several real analysis classes and i needed a few theorums that i couldnt find or remember. i came across this link, and everything was arranged so well that not only was it easy to find the things i wanted, but they were well described. bravo!

- 1314. eric schaertl
- 1313. Po Chávez
- 1312. Nic Davirro
- 1311. Rouzbeh Dargahi
- 1310. Dante Capezzani
- 1309. Ramón Chacón
- 1308. Alcides L. Diaz
- Hi. This looks like a good place for me to "brush-up" on real analysis. I'll be giving you commemts as I study your online text book. Thanks.

- 1307. Karen Shelton
- 1306. russell warren
- This is a very clever method for introducing the basic concepts of real analysis. The best feature is the nteractive capability for getting immediate feedback on the exercises. It would be good to see extensions to a somewhat higher level, eg. Radon-Nikodym theorem, Daniell

- 1305. Barry Grelick
- I went to U. of Michigan and received a BA in math in 1971. I wish this aid had been available when I was taking Real Analysis. We used the textbook Sets, Sequences and Mappings which your site seems to follow, coincidentally, though the proofs are much easier to follow in your site.I would be interested to learn of developments/additions to this site.

- 1304. benbara
- 1303. clevinson
- 1302. Steve Swenson
- I am using this text as a tutor for my class in analysis, so far, so good.

- 1301. pedro bazan
- 1300. Fernando Cabello-Astolfi
- very many thanks for making publicly available materials like these. it so much improves student´s life quality...

- 1299. tawfik
- I want some analysis

- 1298. Monica Shaw
- This site rocks. Thanks!

- 1297. Kathleen
- 1296. Glenn Blanchard
- 1295. Cesare Vecchiarelli
- 1294. René Mario U.R.S
- Just looking for some theorema and its proff , i don't understand what is my teacher saying ! huh :( Wonderful INTERACTIVE GUIDE !!!

- 1293. Pam Poppe
- 1292. Suda Kunyosying
- 1291. julian comalie
- 1290. rima kalawi
- 1289. Hans Zwart
- 1288. joe namy
- 1287. jan schut
- 1286. Peter K.H. Gragert
- I see that you already implemented Java-applets containing the same idea as I did. http://www.math.utwente.nl/~gragert/Live_Math/Continu.html …Maybe you replace Continu.html by TaylorError.html to see what I mean. Maybe Taylor is useful to you too.

- 1285. munjed samouh
- 1284. Amirul Islam
- 1283. Michael A. Conaway
- I'm just an old guy who has a curiosity about Real Variables.

- 1282. jose Garcia
- 1281. REZWANUR RAHMAN
- It is very nice & excellent book.I have some new thoughts in mathematics.Please inform me how I can explore them internationally.

- 1280. Mike Jackman
- Really appreciate the layout and level of difficulty. Hope it gets completed soon. Geat job.

- 1279. Michael Connolly
- I am taking a first-year graduate course, Introduction to Real Analysis. I am an electrical engineer by training and trade, so a good bit of the material (and a lot of the "mentality") is new to me. I appreciate your well-designed and useful web site, Regards,

- 1278. Roberto Zampa
- 1277. Raman Pall
- This is one nifty piece of jumbalaya! I love it. Analysis meets the Internet... sounds like one of those old Japanese monster flicks. Keep up the good work, g-funk. Catch you on the flipside, Raman

- 1276. Domingo Mejia
- nito@ufm.edu.gt

- 1275. Han Zhang
- I thoroughly enjoyed myself browsing thru this page. :-)

- 1274. Christopher H. de Castro
- 1273. Ira Ford Jr
- 1272. Kostas Arvanitis
- I think that it is a very usefull site for my class

- 1271. Victoria Franklin
- Needed exact definition of cardinal numbers for thesis on "Cantor and Infinity" (Logic,Philosophy)-.

- 1270. Giorgio
- 1269. Muhammad Salim Ghauri
- Dear sir, you are requested to please guide us how we can obtain the material on the following topics on real analysis. (1) sequences and series along with their convergence and divergence. (2) upper and lower limits of sequences and series. Muhammad qasim

- 1268. Al Mannon
- I am really enjoying your site. I have been a visitor many times in the last 6 months, so I thought it was time I signed your guest book. I am sure I will return many times in the near future.

- 1267. Humebrto Mejia
- Just invstigating trancendental numbers nad other analysis topics

- 1266. Dario Fantoni
- 1265. SHYAM CHANDRA AGNAJA
- Sir, i have read this book and found nice one,if you have any other such books on subject : algebra,topology and complex variable , pease inform me at given e-mail address. thanking you, shyam

- 1264. rob
- 1263. Carlos Martínez
- EXCELENT

- 1262. Marie LeBlanc
- 1261. Steve Conn
- I'm a high school teacher in Michigan U.S. So far, I've gone through chapters 1 - 3 pretty thorougly. Looks great!

- 1260. Francesco Pattarin
- I have just started using IRA, but I have already found it easy to use, convenient and funnier than most textbooks on real analysis I have looked at. I think it is great for all non-mathematicians (like me) that are approaching the topic for the first tme. Thanks for having made it available to everybody on the Web.

- 1259. Peter Pringle
- Nice to see a good site with lots of questions and answers useful for revision.

- 1258. Shawn Edney
- 1257. Carlini
- Very good! Please finish Metric Spaces. Best regards, Carlini

- 1256. Robert Baker
- Helpful site!!

- 1255. Tad McArdle
- No mathematician I, just came across this by chance, but the graphics are terrific.

- 1254. Antonio Barata
- 1253. Leonard Huang
- This is a pretty good website for students studying real analysis.

- 1252. Price Kagey
- Very nicely thought out and done. I had some wonderful instructors , but wish I had a tool like this when in graduate school (dark ages ). Price K.

- 1251. steve conn
- 1250. Michael Skibsted
- 1249. Majid Basheor
- 1248. Hector Luis Mojica
- Your website was a good complement for my class. Thanks for the help.

- 1247. Vito Bica
- At first glance, looks like a very interesting site.

- 1246. Steve North
- Quite obviously a good resource for beginning undergrad students, as well as for the motivated final year A-level student.

- 1245. Georgina
- I am studying HNC in Software Engineering and one of the modules is Discrete Maths. I have been looking for weeks for help and your site has everything I have be looking for. Thanks, maybe now I will begin to understand Set Theory.

- 1244. Catrin Süssmann
- I was just looking after some help with my problem in math-homework but I dont know if I'm gonna find something about "Logarithmus"!? But your website is kinda cool!

- 1243. Paul D. Humke
- 1242. Rita Rossetto
- I'm an Italian maths teacher, female. This e-book is wonderful. Thank you

- 1241. Deb Mohanty
- A very informative website.

- 1240. Trent Lalonde
- Thanks for all the help with my paper on Lebesgue Integration and Measure. Your book reads a lot like Royden.

- 1239. David Baldwin
- 1238. m Jacobs
- 1237. Bruce A. Johnson
- 1236. dhara
- I am a student of b.sc first year and i tried to gain knowledge about graph theory and its application and partial order lattice and zorns lemma but i was unable to do so sir please give me the possible knowledge of above soon i will be waiting for it please Email to me.

- 1235. Esthela DelToro
- 1234. Bahman Ghavimi
- Great job! I hope to see a similar text for abstract algebra soon. Thank you.

- 1233. Frank Fritzsche
- very good!

- 1232. Joao Carvalho
- very nice. different kind of work.

- 1231. SERGIO P. OSSE:
- 1230. Alfonso Fenandez-Vazquez
- 1229. Corinna
- 1228. Bruce A. Johnson
- 1227. lee cohen
- 1226. KORNER
- Let $E$ be a non measurable subset of $[0,1]$ and $I_{E}$ its indicator function If $f=I_{E}-I_{[0,1]\setminus E}$ then $f$ is not measurable but $|f|=1$ is. An interesting and well thought out project.

- 1225. michael p
- 1224. David Hilbourne
- A very useful site for my IB Higher Level students. Keep up the good work.

- 1223. marek bzikot
- Dear Sir/Madam, I'm writing to let you know that we've put a link to your site on ours (www.chalkface.net). The link is in the Mathematics Resources section (from the homepage, follow Resources Mathematics). Chalkface provides teachers with educational resources as well as extensive lifestyle features, such as an on-line travel service. Our management contains several experienced schoolteachers. Part of our mission in getting teachers on the Internet is to overcome constant re-inventing the wheel; making your site more prominent in this way will enable more effective use of resources. I would like to ask you, therefore, to consider mentioning Chalkface in your mailings to schools and teachers. You may wish to consider other mutual promotions with Chalkface, or you might be interested in advertising on our site. Do contact me for further information, or to discuss any of these matters. Yours, Marek Bzikot, Director (marek@chalkface.net)

- 1222. laurentiu Stefan
- 1221. eallman
- 1220. T.Gruenwald
- these sites are mostly better than our school-books. have you ever thought about a german version?

- 1219. Haley Petersen
- 1218. Ana Maria
- Curso muy interesante

- 1217. Daniel Wolff
- 1216. Sauli Hartikainen
- 1215. P. de Vries
- 1214. AGELAKIS NIKOS
- 1213. David B. Hedrick
- This looks like just what I need to further my studies. A very impressive etext. The glossary is wonderful.

- 1212. James Dagoo
- Interactive Real Analysis is awesome. James

- 1211. Aaron D. Gould
- 1210. pramod kulkarni
- 1209. Chris Stuart
- Free, on-line books are a wonderful thing! Keep up the good work! Chris Stuart, Eastern NM Univ.

- 1208. Fernando Novoa
- 1207. Yalçin
- 1206. Jan Horton
- 1205. Dr. Amnon H. Eden
- 1204. kentril white
- 1203. Wouter
- I'm a student of physics at the university of Groningen, The Netherlands. Your explanation of the Lebesgue-integral was very useful to me. Thank you.

- 1202. Eric Hutchinson
- 1201. Jorge Romero
- 1200. Mike Handy
- Just checking it out!

- 1199. abdulkadir
- 1198. Bob kwiencien
- I wish that there was something like this when I was in college. This helps make many difficult somewhat intuitive.

- 1197. Philip Larso
- 1196. Davina K. Tompkins
- I need HELP!!!!! I seem to be the only one in my class holding out hope of passing the course, but I have to get my homework done. Problem is, I have barely a clue on how to go about it!!! HELP, I'm sinking.

- 1195. Dietmar Logoz
- Fantastic textbook - better than many printed (German) books I have looked at. Question: Can you please add the Dedekind Cut proof? Kind regards, Dietmar

- 1194. Jake
- : I have not yet researched this site, but I am looking forward to utilizing it! I am currently studying real anaylsis and I could use the extra resource!

- 1193. Lance Hemlow
- Unreal. That's all I can say!! I wish I had this site when I sweat through all these calculations! Nice work. Lance E. Hemlow, Raritan Valley Community College

- 1192. Michael Herschorn
- 1191. Rick Nelson
- Very impressive - a clear and easy to follow presentation. Well done! I look forward to the download version.

- 1190. Stefania Latorraca
- 1189. Tania Robbins
- 1188. Sebastian Anand A
- 1187. A S Jones
- 1186. Jim Moore
- 1185. Ana Garcia
- Muy bueno

- 1184. Rui Santos
- 1183. Ralph Shelton
- Good site (sight) for us self-proclaimed intellectuals. Thanks for your energies, Ralph

- 1182. Cristiano Dane'
- Interesting and well done. Thank you.

- 1181. Sue MARTIN
- NSW HSC Online (http://hsc.csu.edu.au/) is a co-operative venture between the New South Wales Department of Education and Training and Charles Sturt University. It is a web site to support students and teachers involved in the NSW Higher School Certificate and it harnesses the resources of the Internet to support curriculum in a focused manner. As your web site has been identified as one that would be of educational value to the teachers and students in NSW schools, we wish to provide a direct link to it in Mathematics.

**permission granted** - 1180. Brian Davi
- 1179. Salim S. Akl
- A very interesting and helpful site. Keep up the good work, GOD Bless you

- 1178. Jozef Van Remoortere
- 1177. Lisa Celec
- I am having trouble finding anything to do with a function that is additive(preferabley a proof). I was wondering if I'm not looking in the right chapter. Do you have any good resources that you may suggest. Thank You

- 1176. Lisa Harrison
- This is a great site - the examples are much more explanatory than the ones in my Advanced Calculus book.

- 1175. anuar
- The explanation is easy to understand. I'm very interested in real analysis although sometime take several days to finish a page.

- 1174. Esther Heck
- 1173. rabeya basu
- searching for book of real analysis

- 1172. Peter Joseph
- 1171. kenneth spencer
- 1170. robb todd
- 1169. Lars Roland
- Great project you have here.

- 1168. Kelly Russin
- I am a senior in college taking advanced calculus and I am struggling.

- 1167. Bhavana
- I appreciate your website.

- 1166. K B Laursen
- I am in charge of our large analysis course (app. 400 students) and am interested in everything that makes this kind of material relevantly available on the WWW. So I shall take a very serious look at your work!

- 1165. Jason VanBilliard
- 1164. mahmut
- 1163. cutimanco
- 1162. Lisa Harrison
- 1161. Thomas Richardson
- Nice to find such great online resources.

- 1160. bud
- 1159. Diogo Bessam
- Very good work indeed. Fantastic for self-learning!

- 1158. abhijeet throat
- Sir, book is very good .

- 1157. Melanie McLeod
- 1156. Brandon
- 1155. Phil Mendelsohn
- I find you can't really go over Analysis too many times or different ways. My memory being like an open set, (i.e., it does not contain all it's limit points!), this is a nice reference to remembmer.

- 1154. Ann Orr
- Really neat website. Has been awhile since I looked at continuous functions and the rules that apply. Helped me convince a colleage not to waste time trying to figure out the function for an inventory process because his graph was illistrating a line that was discontinuos.

- 1153. Luis Belcredi
- 1152. sanjay kumar
- 1151. marson early
- What I have read so far the book is excellent. Is there any hard copy available for sale I would be interested in buying a copy

- 1150. Fiona Case
- 1149. h.r.vishki
- 1148. Gerzson Peter
- Very Nice ! So, Congratulation !

- 1147. Christopher Comparini
- 1146. Dr.Hailiza Kamarulhaili
- 1145. gokhan kartal
- 1144. chris fraser
- 1143. erwin susanto
- Good morning. I want news/information about jurnal real analysis after 1995

- 1142. John M Fox
- 1141. Ben Torbenson
- 1140. Luis Belcredi
- 1139. Derrick Bertrand
- This type of site if wonderful. Years ago, if the internet were at this stage and this site was readily available, I would have been all over it. Is there a real analysis site that starts with a theorem and definitions and works it way up in a sequence to more dificult ones? I am working on my B.S. in mathematics and would like online sources to reference. Thank you, Derrick Bertrand, University of North Texas, student

- 1138. arastu
- Nicely done. I don't want to blaspheme but in terms of the clarity of conveying information its a step up from Kolmogorov and Fomin.

- 1137. robb todd
- 1136. Marcia Edson
- 1135. ncracho
- I still have to browse the contents!

- 1134. Luis Trejo
- 1133. Bonnie Babbitt
- This is a great help---thank you so much!

- 1132. cvrv
- This site is quite good but I was particularly interseted in finding an inteactive 'math/geo plot that allowed me to click and change a line to see the changes in the associated linear equation. Can you point me in the right direction? (Regards... Charlie)

- 1131. Isami Yamada
- 1130. Stefania Latorraca
- 1129. koksal atik
- As a graduate student at Bosphorus University in Istanbul, i make use of your online real analysis textbook extensively. I thank you for that! I think, what u 've done here is a very helpful GIFT for the students who are in trouble with Real Analysis here :))) (with an industrial engineering background). I really very appreciate your work here. WELL DONE!!!

- 1128. Daniel Wolff
- 1127. Lovejoy Nyandoro
- 1126. Ashley
- What is the sum of all the digits of all the (+) integers from 1 to 1,000,000 ?

- 1125. Bill Watson
- 1124. M. Sabato
- You have made a usually obtuse topic unusually accessible. Much thanks.

- 1123. Sipho. J. Magagula
- Real Ananlysis is a interesting subject of mathematics. Can you please give more exsercises.

- 1122. Catherine Beneteau
- I'm using this interactive text to teach real analysis in an independent study setting. So far, I am very happy with the text, which seems accessible and yet suitably challenging. The problems and solutions worked into the text are very helpful.

- 1121. Filipe R. Fonseca
- 1120. Colin Bennett
- 1119. Raymond Chia
- Hello, I would like to introduce myself : I am a teacher and currently doing my first degree course in Mathematics throught distance learning. I am taking the Real Analysis course for this semester and I face difficulty to understand the way how to proof the equations in this course. I hope you can help me.Thank you!

Question :1) Let S subset of Q(rational number).Is it possible for S to be an interval? Question : 2) Let a , x and y are elements of R (Real number) which satisfied

`a =< x =< a + y/n`for all n element of N. Proof that x = a - 1118. jason nobles
- 1117. J.F. Kaashoek
- 1116. Joseph Khalil
- Thank you for a great book. This book is very well written and makes hard material simple to understand.

- 1115. Doug Carlson
- 1114. M. Sabato
- IRA is a very accessible treatment of a usually obtuse topic. However, I can't seem to find here anything on the Diagonalization of the Cantor Set. Where do I look?

- 1113. Andrejs Treibergs
- 1112. Fred Schifando
- 1109. Diane Lanigan
- I desperately need to see completed proofs regarding Cauchy sequences and the real number axioms! Please?!

- 1108. anthony segers
- This may be the best mathematics site on the net! SERIOUSLY! Please consider including complex analysis.

- 1107. ammar
- my name's Ammar Osman study on faculty of math in Khartoum university in Sudan and I need the some books for math one of them the system analsyis and other.

- 1106. Tracy Payne
- 1105. Ken
- 1104. Pietro
- 1103. Chelsea C.
- 1102. ara Roark
- Interactive Real Analysis was very helpful. I would have liked to see information about the "characteristic function".

- 1101. John Godsill
- Thanks for making this available. It is an extremely valuable source for those of us studying for maths degrees. Is there a corresponding site for groups or combinatorics?

- 1100. LAWSON,MARCUS
- THIS LOOKS VERY GOOD. I HAVE NOT HAD TIME TO USE IT YET.

- 1099. Abundio Adriano
- 1098. jill nesbitt
- 1097. Sergei
- 1096. christopher Bognar
- 1095. S.H. Kulkarni
- 1094. S.Krishnakumar
- 1093. T.Barnes
- 1092. S.krishnakumar
- After I refered this book, I will comment.

- 1091. Igino Contrafatto
- 1090. B.Zandstra
- 1089. marson early
- 1088. Ryan
- 1087. Feral Child
- 1086. m early
- 1085. Heidy Cecilia Chavira Jara
- 1084. Juan Viramontes M.
- 1083. Dr. Jeanne Mortinson
- I am looking for any online courses in real analysis or modern geometry. Are there any such courses out there? Thanks.

- 1082. rolf johanson
- 1081. Jacquelyne
- 1080. EVELYN VARNER
- 1079. Diogo Bessam
- I wish to congratulate the ones involved on the fine work. Very nice indeed. I also want to ask you to carry out with the rest of the material you have on metric spaces and on the issues that are incomplete that is not finished yet. Congratulations!

- 1078. carol chan:
- This has been a great help to me, thanks.

- 1077. Cavenecia
- 1076. Robert C. Middleton
- I think that it is commendable that you have developed a Web course in mathematics at this level and are willing to share it with the world.

- 1075. Steven Wu
- I want to get any guest books. Thank you!

- 1074. Pat McCormick
- 1073. mahmut kocak
- 1072. Cecilia Medeiros
- 1071. Steven Wu
- 1070. kristin wilke
- 1069. Anders Henningsson
- 1068. michael
- 1067. al massanova
- 1066. kimikpyo
- 1065. srikanth
- Sorry, this is my first visit.

- 1064. Bennyfer Bridgewater
- 1063. Julieta Rangel
- This interactive book has been so helpful. Thank you. I just hope that the last chapters get updated soon so we can use them.

- 1062. David
- Great Site. Comprehensive treatment, but not to hard to understand (even for non-mathematicians).

- 1061. Roy Klesczewski
- 1060. Okondu J. Chuks
- I was at your site for a while and I really found it helpful for my class assignment. Keep it alive. Thank you.

- 1059. Okondu J. Chuks
- I hope to make this site a companion. Comments may come when I go through your site

- 1058. Renee Archoleka.
- Very nice and organized work. It is very helpful

- 1057. Rick Lutes
- 1056. no name
- 1055. Gshansen
- Wow. Sorry... WOW! I make models and displays, and this is a kick. It is also time I learned this for real. Thank you all. Be at peace.

- 1054. No name
- 1053. Herve Chappe
- Excellent and very helpful ! Please finish last chapters...

- 1052. chris
- Very nice job, great resource. Well laid out

- 1051. Javier Rivera
- Excellent!!

- 1050. J. Munoz, Jr.
- 1049. Don Coleman
- 1048. Ryan O'Donnell
- Excellent resource. Has helped me to understand Analysis II at Michigan State University much better. Thanks a lot.

- 1047. Silalahi
- I want to learn about Real Analysis

- 1046. miyuki maruyama
- 1045. Ugur Kuru
- Thanks for this style of web usage

- 1044. Pasquale Scarano
- 1043. sroor asaad
- 1042. Vera Horvath:
- This website is a very useful one for teachers, really great help! Thank you.

- 1041. jason wu
- The hyper-links for the questions in 7.4 do not work. Could you please repair them? Thanks.

- 1040. Marlon
- Explorations of dynamical systems: If the derivative of a dynamical system equation is between -1 and 1 at an equilibrium value (fixed point) of the system, the fixed point is an attractor; if the derivative of the dynamical system equation is greater than 1 or less than -1 at a fixed point of the system, the fixed point repels. What initial values are attracted to an attractor? What happens at fixed points where the derivative is +1?

- 1039. Emerson Rildo
- 1038. G.S. Wei
- 1037. anthony wiggins
- 1036. Lav Mehta
- 1035. hellou azmi
- 1034. Marc Shaner
- 1033. Jack
- The online RA book is really cool. It's great if I can have a copy home.

- 1032. M.Kasigwa
- Thank you for the good work

- 1031. inga safovich
- 1030. Luis Belcredi
- 1029. sigoldiber
- 1028. Helen McCulley
- 1027. Ulrich
- 1026. john deangelis
- Excellent idea! Very Informative.

- 1025. Adam
- Good stuff- easy to use and understand. This site helps to take away any feelings of intimidation by the subject matter -a.n.

- 1024. Naveenkk
- 1023. tom
- One of the most detailed and interesting sites for real analysis that I've seen.

- 1022. Istvan Diosi
- Hi you there, I'm mathematician and I live in Szeged (Hungary). If you travel in functional analysis you have to know this place, because here teached F. Riesz, L. Fejer and B. N. Szokefalvy. The idea of your site is very interesting and useable and I will mention to my students. Thanks for your site. Have a nice weekend. Istvan Diosi, Szofi, Algorithmic Research Inc. (Szeged, Hungary)

- 1021. John Vassey
- 1020. Mehul Mehta
- I am in the field of teaching mathematics. I am conducting classes for graduate students. I can help u in developing proof, examples, related to cut theory topology, sequence, series, power series. I am also interested in abstract algebra. Let me know, from Mehul Mehta

- 1019. Tim Calhoon
- 1018. Cherng-Yih Yu
- 1017. Len Lavoie
- 1016. Tony Gaines
- Great idea! I've been looking for something like this for about seven years

- 1015. steve easton
- I can use this to prepare for my intro to analysis class. Thank You

- 1014. Paul Horth
- 1013. Carlos Carlini
- Very good. Please finish the chapters!

- 1012. Lucio Benaglia
- 1011. Brock Stewart
- 1010. Nadeem Haider
- Sorry, I am just about to visit the site

- 1009. Andr? Novroth
- 1008. declan Walsh
- 1007. Domingo Mejia
- I like the way you do the proofs and the treatment of the cantor set. Thanks for all this.

- 1006. Edward A. Moore
- This is my first experience with this site. What I see is that this material looks to be very helpful for my needs.

- 1005. Winfield A. Ihlow
- I stumbled upon this website by accident while I was looking for something else. I congratulate you on what looks like an excellent project and excellent service to teachers and students.

- 1004. Milan Lukic
- 1003. Robert G. Kavanagh
- 1002. Bob Fennell
- 1001. B.P.Lamichhane
- It is really interesting and helpful to find so many things of mathematics in the web.

- 1000. Helen McCulley
- 999. Gerald (Jerry) Hyde
- None so far. I will let you know if I have any. J

**You are the 1000ths person to sign our guest book - no prize, though, sorry; bgw** - 998. Matthew Arvanitis
- 997. CARLO CAROSELLI
- thanks

- 996. Adithya Yaga
- 995. J. A. Ziegler
- 994. Frank van Dijk
- 993. Hien Tran
- 992. morton brown
- 991. douglas williamson
- 990. Olusola Fadero
- This resource is proving a real help with the math bits of my masters course. Thanks who ever put it online.

- 989. Stephen C. Webb
- Tripped across your excellent presentation and just wanted to let you know that I'm very impressed with the quality and ease of use of the presentation. One quick note: In the Map, the use of Dark green for the topic areas with black lettering makes it quite difficult to read. Keep up the good work, the availability of sites like this can do nothing less than propogate interest (and hopefully excitement) in our discipline. Kind regards, S.C.Webb -- Sr. Research Engineer Member IEEE, AMS

- 988. William Reade
- Ooh, lovely. No, seriously, it's a relief to have a reasonably clear set of notes...

- 987. Ken Barrow
- 986. Frank van Dijk
- Clear way of explaining. Being a teacher of some of these topics myself, I would like to see the proofs that are said to be put later.

- 985. CRISTIANA MAMMANA
- I am very interested. I am Professor at MACERATA University (Italy). Send me mere informations and News. Cristiana

- 984. Thomas Brotz
- This is my first visit to this site. So far so good :-)

- 983. brianwallace
- excellent

- 982. Joe Sturtevant
- 981. Erin Rozman
- I haven't started looking at the material just yet, but this looks like a great site!

- 980. ART MARK
- Enjoyed visiting this site. Good review, looking forward to a visit whwn you complete the last sections.

- 979. Gabriel Fagan
- Dear Professor Wachsmut, I just want to congratulate you on your excellent Interactive Real Analysis Course. It really demonstrates the potential power of the WEB as a pedagogic tool. Just two questions: do you intend to develop the site further, e.g. by completing the chapters of sequences of functions and metric spaces? What 'hardcopy' textbook would you recommend to accompany the interactive course? Once again, congratulations. Regards G. Fagan Frankfurt am Main

- 978. KevinThompson
- This is really cool. Keep it up.

- 977. Sarah Kasten
- 976. B. McKeown
- 975. James Quinlan
- just a comment about the main page, your scrolling script is red on blue, not a good combination, ask a graphic designer the best is white background, and black text keep it simple is keeping it real red on blue or vice versa is to hard on the eyes

- 974. Jorge Salvador Montes
- good

- 973. mehulmehta
- I am in the teaching field. I am new user for your site

- 972. James Quinlan
- I love your project. Please complete it. Some of the sections have broken links, or where you have examples there are none. In particular Riemann-Stieljes integral section is not done. I want more!!!!

- 971. Miguel Angel Acebo Visanzay
- very good

- 970. marco garcia
- nice site, follows Interactive Real Analysis by Micheal Reed very well, finish it up please

- 969. Miguel Angel Acebo Visanzay
- 968. beth wehrmeyer
- Thank you so much for having this site it is helping me graduate.

- 967. Peter L
- very impressive site. Not only to the correctiness of the information provided about real analysis but also the use of java applet to show the graphs. Especially the applet help me understand many important concept, thanks :)

- 966. James Morgan
- 965. Karen Mobley
- I came to this webiste to find help with clusters. However, I couldn't find this particular topic. If it is included, will someone please tell me where? Otherwise, is there any way someone could help me? Thanks!

- 964. Kristin Webster
- 963. Dr. Roger Stedman
- Well constructed site. Enlightening for an sub-amateur mathemetician even! I am interested in modelling physiological systems using continuous algorythms as opposed to the traditional discrete time slicing method. Any pointers to other rescources would be welcome.

- 962. David O'Connell
- 961. Wynand Bezuidenhout
- Hi I am a third year student in math in South Africa and I would have liked to see some of your stuff about metric space (i have to do a project on metric spaces)

- 960. Sandy Austin
- Very informative site and much appreciated.

- 959. Ramadhar Sharma
- 958. dipak banik
- a good book and easy to follow

- 957. Jeffrey W. Collins
- 956. Armand Principato
- 955. Stanley
- This looks like a mighty handy site. I find it helpful to hear more than one voice on any subject, but my school library has nothing to even approach Spivak. I hope to revisit you here. Thanks

- 954. Vincent Tomasino
- 953. James Quinlan
- Get idea!

- 952. Alan Roscow
- This is a great resource for someone like me who is interested in math, but never had the time to take advanced math courses like real analysis.

- 951. Wim Groeneveld
- 950. C.C.
- 949. Paul Fischer
- 948. william scott
- 947. Kelvin
- 946. Weerd
- 945. Yong
- 944. betty clifford
- I need help with a problem. Let a,b,c, and d be any real numbers such that a is less than b and c is less than d. Prove that [a,b] is equivalent to [c,d]. this is taken from Introduction to Analysis, 5th edition, by Edward Gaughan. The text uses the function: f(x) = (x-a)/(b-a) is a 1 - 1 function from [a,b] onto [0,1]. Why? Thank you for your time.

- 943. Daniel Greisokh
- I am a student in an Introduction to Real Analysis class. I have found this tutorial VERY helpful! The tutorial is well written and explains some concepts better than the book that I am using in class, "Analysis, with an Introduction to Proof" by Steven R. Lay. The glossary is especially helpful. I have several suggestions that I have noted thus far: -consider including a section on logic, namely truth tables relating a statement to its contrapositive, inverse to converse -consider including an illustration of de Morgan's Laws. Being a novice, it took me some time to logic through the theorem. An illustration would have been helpful. I will post suggestions in the guest book as I plow through real analysis. Thank you for making this tutorial, Dan :-)

- 942. Warne
- 941. Veronica Estrada
- 940. Rusty Mawk
- 939. M. K.
- This is, absolutely, the most incredible mathematics website I have ever had the good fortune to happen upon. Thank you for the effort it must have taken to build this!

- 938. C.Y. Chung
- It seems there are lots of sections unfinished yet. Can you recommend some reference books to me to study this interactive real analysis course ?

- 937. Sigmund K. Andresen
- 936. EL MAGHAT omar
- please in the case of any new subject in the domaine of mathematics i want to know it as soon as possible. thank you.

- 935. Neil Whited
- 934. Diane Austin
- Thanks for a good place to look up forgotten analysis! It would be nice to have a search function for your site.

- 933. Hong Hue Ha
- Hi I'm a student at McMaster University. I would like to say that this site has been essential and beneficial to me in my analysis courses. Thanks.

- 932. Whitney Riddle
- 931. dipak banik
- 930. Aaron
- I am a student at the University of Washington, currently taking the introductory calculus series. I was wondering if there is a similar online textbook for several variable calculus. Thanks, Aaron

- 929. Thomas Cresswell
- 928. gregory johnson
- great math site. must get the riemann-stieltjes integral working

- 927. J. Howard
- 926. Rik Jamieson
- 925. Manuel de Jesus Rosa
- 924. Joe Miller
- Please keep this site up and running. I just discovered it and am using it to help study. Thanks.

- 923. Mandy McMillen
- This is a very useful page. My professor assigned us to research real analysis on the internet, and this was the best page I found.

- 922. Cristi?n Pinto Rojas
- Wonderful idea. I found the internet rather boring until I got into this site. You have a great piece of material here. I'll keep coming. Greetings from Chile

- 921. Suzanne Munitz
- 920. Jason Doctor
- I haven't had a chance to review the whole course, but am impressed with the use of graphic examples.

- 919. Manuel Cruz
- Congratulations.

- 918. Daria Sharman
- I was studying for a final in my analysis course and your page helped me more than the text ever had! Thanks for the real and understandable lay out!

- 917. Clement
- Informative.

- 916. sultan, Mujahid
- The site gave me efficient and pertinent information about sets and relations.

- 915. William Varhanik
- 914. Shawna M. McElroy
- 913. bulent guven
- 912. Dennis E. Turpin
- interesting site, very informative

- 911. Stanley Cohen
- 910. Irtaza Barlas
- Wonderful effort to put up this beautiful and most useful site. Kindly convey my appreciation to the compilers of the book.

- 909. Rae Deeley
- Have just been updating the link with your web pages that appear on the pages which direct students as to where they might get help for the HSC (NSW) exams. Still looking pretty good resources for our high school students. Thanks.

- 908. Jorge Salvador
- Very Good

- 907. patricia r mangubat
- 906. Jay Cooke
- 905. Vachenauer, Peter
- we also are working on an multimedial course on Integration here in Munich at the Technische Universitaet. our link http://www-hm.tu-muenchen.de/integration

- 904. Dennis E. Turpin
- 903. Rishi
- Im doing M.Sc Mathematics along with architecture.This is a great site for reference.

- 902. Giuseppe Bertoni
- I think that's a very good product;however how can i get a version about analysis of function of multiple variables? Can i have a free copy of your version of the present "Interactive Real Analysis"? Thank you very much.

- 901. michael sullivan
- I was very impressed with the quality of the presentations, both in visual content and mathematical accuracy.

- 900. Ray
- That Beavis and Butthead sound you have there expresses my thoughts quite accurately--I loved it! Are there any others hidden in there? ;)

- 899. Alecia Whiteland
- 898. Claude Lemaire
- I'm a teacher in engeneering school in France Is it possible to obtain your applets java I want to use them in classroom but without connecting to internet, only in local use. Thanks CL :-)

- 897. darren nicholls
- 896. Joe Sturtevant
- 895. A. Kirfoot
- Thank You For taking the trouble to put this on the web.

- 894. Mark Mars
- 893. R. Dalbey
- Interesting, informative presentations. This is my second visit and I'll be back.

- 892. Alessio Lomuscio
- Great!

- 891. HAPPY PARAMA
- I come from Indonesia. I'm an university student. I study in GMU.I haven't see about the real analysis yet, 'cause I'm only looking for math's web. But I think it will be interesting if sometimes I check it out.Because I think it will be help me so much in my study.

- 890. Atul Razdan
- I have just started. It will take me some time to comment upon.

- 889. Jenni Huovinen
- Thank you very much! This site has been a great help to me.

- 888. Mathias Lindgren
- Thanks for an superb site. / Mathias from Sweden

- 887. Guy R Corrigall
- Only discovered your site today (16 October 1999). Am very interested in online math courses, especially in algebra/group theory.

- 886. Danae Romrell
- Great Resource!!! Thanks!

- 885. edmeyer
- wonderful. just the thing to get a 13 year old interested in mathematics, (and,maybe his teacher as well) just about the time he's beginning to check out...and doze off.

- 884. Tara Stanton
- I hope this can help with analaysis. So far it seems real helpful.

- 883. Mark
- 882. Mark Milodragovich
- 881. A.Kirfoot
- Thank you for making available pages to enable me to understand. I never went to school.

- 880. Harry Hirsch
- Outstanding work. It may even be that you are opening the door for a new career for me. Thanks a lot.

- 879. Amanda Dondelinger
- I think that this is a great idea. It's helpful to have other sources aside from my text for class. One thing that I still have problems with, though, is induction. Perhaps more examples would be good-- especially with recursive formulas!!! Thank you!

- 878. Noel Buckley
- Excellent

- 877. Randall
- 876. kwety
- 875. Hal Voepel
- Thought you'd like to know that under your definitions for Even and Odd numbers: i.e. E={x:x=2n for all n in N} and O={x:x=2n+1 for all n in N}. The union is N-{1}, not N as you have indicated. You may want to redefine Odd numbers to: O={x:x=2n-1 for all n in N}. That way the union will be N.

**(Fixed - thanks; bgw)**P.S. I really like your web site. Thanks. - 874. Don Speray
- It's such a pleasure to see what motivated educators *actually* do with the Web. I wish this had been around 25 years ago when I went through Apostol (which I'm doing again, in early retirement, for fun!). Although your site is improving/changing, would you consider providing a single-file downloadable version, say, a periodic snapshot for off-line study? Interestingly, your guestbook is a source of other sites of interest. Thanks!

- 873. Cameron Tareen
- 872. aristide vetere rossi
- 871. Young B. Yang
- Real good! I like it. Thank you for your tremendous job.

- 870. Ivan Cnop
- Am involved in a program for writing math under ICT in Flemish Brussels univ (VUB) using Mathematica

- 869. mike
- great site man!

- 868. Karen Kurdziel
- 867. R.L. Ward
- 866. cassie salmom
- On the proof : There is a pos real # x such that x^2 = 2 Why do you let delta = 2-s^2/2s + 1 ? I am working on the same proof of x^=p. Please help me if you can. Thanks

- 865. banasiak
- 864. David L. Windrick
- Great stuff. Looking forward to the completion of the unfinished portions.

- 863. Daniel Annweiler
- 862. Erik Blasch
- I really appreciate the web page. Haven taken the class many years ago, I have found your references very helpful when I am at work and the books are at home. Since SH funded the program, I commend them and have suggested the site to others. Erik Blasch, Capt., USAF; AFRL/Sensors Directorate

- 861. Natashia Boland
- Great stuff!! Thanks! Lots of content and very nice looking

- 860. no name
- Comment removed as per user request

- 859. Bobby Wattimena
- Eventhough it's not complete yet, this page is very helpfull for me to understand real analysis, especially for me who will take recursive method in economic dynamics.

- 858. fahimi ibrahim
- 857. Tim
- 856. james kahelin
- are there any web sites that teach logic including things like contra-positive. my experience taking real analysis is that you must be fluent with: there exists, such that, for all, the negations of the above, induction

- 855. Paul Ovalle
- Thanks for a great site! I hope that you are able to build more sites like this one with other undergraduate math subjects?

- 854. kable gilliland
- 853. ASP
- Thanks!! This wed site was very helpful.

- 852. Brian Keel
- 851. Rob Weiher
- 850. Michael Watry
- an incredibly thorough and wonderfully organized website devoted to real analysis; I found a large portion of references that I was looking for

- 849. ernest baynard
- Really impressive. This is tremendously helpful in my quest to relearn -- or perhaps learn-- things I wished I had learned earlier. I am a lawyer who pretends he is a mathematician and who fools no one. Nonetheless, it's fun. Thank you very much for your page(s).

- 848. Rich Kaufman
- I just wanted to let you know that I think that I spotted an error on your webpage: I think that the exponent for b in the second to last term of the binomial theorem should be 1 less than the b exponent for the last term. (i.e ....b^n-1 + b^n)

**(Fixed - thanks; bgw)**Sincerely, Rich Kaufman - 847. Sheri
- Very helpful. I'm using it as preparation for a Dynamical Systems Theory class.

- 846. Helen Cook
- Excellent! (except for the pages not yet written :) ). The Weierstrass function is.. fascinating. I would really like to see a proof of why it is not differentiable. There should be more online books like this around. It would be cool to be able to download all the html in one big tarball for us stuck with a dial-up connection. Thanks for providing a great service.

- 845. joseph guse
- This is excellent. Exactly what I was looking for. I've shopping around lately for an intro analysis text to congeal my piece-meal learning of the subject thusfar. The presentation is better than in any paper book i've looked at. Looking forward to your completion of the final chapters.

- 844. Van
- I think this is an excellent resource for students. I am trying to bone up on this area of mathematics in order to prepare for graduate study. it's a perfect reference tool.

- 843. junek
- It looks nice. Do you have estimartions by the students?

- 842. David W. Hutchison
- Nice work.

- 841. Rico Argentati
- This is a great review of the material. I especially like the examples and the nice enumeration of concepts in an organized way - crieria for convergence of theory for example.

- 840. Grace Schoonover
- 839. dang van duong
- 838. Thomas Oehlcke
- We are a class of 17 years old pupils at a technical college in southern Germany. We enjoy your program very much. May be there is a mistake in the answer to the examples for differentiable functions. The results of the limits shouldn?t include the variable x any more. No serious problem!

**(Still, was fixed - thanks; bgw)** - 837. walter reuter
- What a great present for all students and teachers. May be there is a little mistake in the definition of uniform continuity. Substitute x for the s or vice versa.

**(Fixed - thanks; bgw)** - 836. Walter Reuter
- Thank you very much! I?ll try to teach real analysis to my German class with your wonderful "internet book". P.S.:Is there any connection to the German Prof Wachsmuth in Bielefeld? Not that I know ....

- 835. Thomas Hayes
- 834. David Adams
- I am using this web sit to study for my qualifying exams for grad school

- 833. Walti
- 832. Belcredi, Luis
- Excelente trabajo, mis felicitaciones. soy profesor de mate?ticas en Uruguay y he consultado los trabajos de ustedes con mucho ?xito. Gracias

- 831. John Hunt
- I'm a graduate student in mathematics and I'm looking for multimedia teaching tools. Regards, and God Bless, John Hunt

- 830. Candida Segarra
- 829. Mohammad Alzohairi
- It is a great work.

- 828. karol hricisak
- 827. David Hurowitz
- I have only taken a few semesters of calculus and a course on applied calculus and differential equations and I have found this site extremely useful. I have bookmarked it and will continue to come back. The interactive quizzes really help in testing understanding so that even a non-math major like myself can understand a fair amount of the material.

- 826. Lewis Royal
- 825. Linda K.
- 824. Calandra R. Tate
- Hello Dr. Wachsmuth, My name is Calandra Tate and I just graduated with a B.S. in Mathematics from Xavier University of Louisiana. I was writing you to commend you on such a wonderful job with the interactive website on real analysis. I had the course this past year during my senior year and it was quite challenging. As a matter of fact, I am retaking it this summer at the University of Maryland College Park to obtain a better grade as a provision for entrance into their Applied Mathematics Ph.d. program. I have only been in the class for a week and already I start to recall the horror of last year. I accidentally searched upon your website and I think its great. I have never seen analysis look so interesting and well explained. Students get a chance to see the whole course outlined and presented in a way that helps you see how things are connected and build on one another. I intend to take full advantage of this website to aide me during my course this summer. And hopefully, I can tough through it and get the higher grade afterall. Thanks again for this magnificent idea! Gratefully, Calandra R. Tate

- 823. Judy Koehnlein
- This is very useful. Thank you for the explanation of the proofs. I'm taking a real analysis course at Ohio State and the professor doesn't help much and the book is so compact that the proofs are incomplete and there are insufficient examples. Thank you and keep it up.

- 822. Alberto Capitani
- Nice service!

- 821. Jonathan Drews
- Your descriptions of measure and the Lebesgue integral are very good. This is precisely the coverage I needed. Keep up the good work!

- 820. Vangelis Haniotakis
- Excellent site, taking students from the bare bones to advanced levels. Still, I would prefer fewer low level resources, and more advanced ones. Moreover, I recommend that you check for broken links and images (e.g. in the Lebesgue integral section) and correct the spelling while you are at it. All in all, a great resource, keep it up.

- 819. Andrew Dunbar
- WOW!!! This is really fantastic, what a great way to learn. I am doing this at 12:50 a.m. and I am not in class. THANKS.

- 818. Candida Segarra
- I really found your page extremely inresting especially since I am preparing a presentation on the Lebesque integral. Your use of color, detail, and graphics was very creative and inviting. Thank you very much. I can't begin to tell you how many hours I've spent at the library and on the internet researching information. When I found this page, I felt so relieved as the information was so clear. Thank you very much.

- 817. Gordon Davis
- 816. Jason Magers
- 815. Aaron Bohannon
- An online textbook is a great idea! The site is constucted very well, also. Although the content seems to lack some details in some places, the benefit of a web-text is that it can constantly be updated and improved. Hopefully, there will be more text-book sites following in other areas.

- 814. C.Thao
- IT definitely cleared some confusion out of me. :) Thankz.

- 813. Ron Dier
- a useful summary of concepts presented in my second year university real analysis course... I've referred some peers to it. Unfortunately, this is on the eve of the course final, but it is worth passing on. Thanx!

- 812. Bob Curry
- I have just found your site and have also just started in a M.Ed. Program in math. I'll send more comments as I get into it.

- 811. Uwe Steinhoff
- Really useful site, thanks.

- 810. Sigrid Heizenreder
- You've created fine lessons in real analysis. I enjoyed very much reading the chapters. Greetings from Germany, Sigrid

- 809. nadiyyah
- 808. Alan Walker
- Bert, A few weeks ago, someone challenged me on the convergent subsequence of sin(n) question. I found your web page after about a week of sleepless nights, but I didn't find the answer on your site. Would you be so kind as to tell me how to do this one? By the way, I would have liked to have had all this stuff on-line when I was at school. Has anyone gone as far as teaching math directly from a web site, w/o a companion text? Thanks, Alan.

- 807. Victor Scheff
- I just discovered the site and it certainly looks good so far. In particular the explanations seem to give an intuitive notion of whats going on rathar than just formal. Thanks, VAS

- 806. Carlos Yocupicio B.
- It's wonderfull find an exciting place like this, I enjoyed a real aventure called Interactive real analysis and I hope you can walk trough of this mathematical fantasy.

- 805. dan nicol
- 804. Cassie Wagner
- 803. Randy Mardak
- I'm a student at Lawrence University in Appleton, Wisconsin. I was stuck on my real analysis homework for about two and a half hours and ready to give up. As a last resort, I went to the internet and happened to come across your online text book. In less then ten minutes I was able to bring everything together and make sense of my homework. I wanted to thank you for this valuable resource and the time you spent to make it available. Thank you again, Randy Mardak

- 802. Ehrlicher, Gerd
- Visited just 4 fun. Enjoyed it. Will be back. Tschuess, Gerd

- 801. jesus
- 800. Tanya F. Lloyd
- This book proved very useful for a project on functions I had to do. It gave me different examples of both continuous and discontinuous functions. However I really would have liked it if the last chapters were available so that I could get more infomation on sequences of functions!

- 799. Lisa
- You have helped me soooooo much! Thank you!

- 798. Marc A. Hill
- I hail from miami fl. , this is an excellent site. hope it is completed soon. if yu need any help, give me a buzz.

- 797. Moti Levy
- Thank you!

- 796. Brandi Holcomb
- This site is very useful for us Advanced Calculus students in Texas!!!

- 795. Jesse Warren
- An excellent site for a student. I especially liked the pull-out proofs and specific examples with graphs. I am currently in my 2nd semester of analysis at the university of Houston at Clear Lake under Dr. Michael Mezzino.

- 794. donna allen
- 793. Tom Birtley
- I have seen only the "Road Map" description of this site but I am impressed by it. I don't know how much I will utilize the material at this point.

- 792. maryannpoppins
- 791. Shurick Makedonsky
- 790. Erica Lewis-Wilson
- 789. Betty Mayfield
- I stumbled upon your site via the MacTutor History of Math Web site. I am giving my Real Analysis students the option of exploring your site as a final project. An interesting idea!

- 788. fadia_azzam
- 787. jim kawa
- Enjoy playng with it. Good mental exercise.

- 786. Hector Aguilar
- 785. Reson Holt,Jr.
- Please send me any information on Carl Freidrich Gauss

- 784. Parker Niemann
- Excellent site.

- 783. Randy Smith
- hey this site is great!!! world class!!!! Keep up the good work!

- 782. Mauro Gaetano
- Outstanding effort. Well presented, easy reading. The pronouciacion of names is such a great idea. Keep up the good work.

- 781. Moumen
- 780. Charles L Davis Jr
- I can not attend school at the moment and hence must continue my mathematics and physics studies at home over the net with the textbooks I have. I have been to other sites that have online classes in a variety of mathematics areas, but this site is really the best I have seen. Fine work and a real asset to the stay-at-home math student. Thanks for the good job.

- 779. Phillip Rhodes
- 778. Quinn Goldstein
- Well done.

- 777. Charles Bush
- 776. Nico du Bois
- It helped me - being not a mathematician - in understanding and handle some math.problems I encountered in my research for the use of a fuzzy logic system in occupational medicine.

- 775. Paul Dyas
- Good site! It has been very useful with my Advanced Analysis class I am taking at Tarleton State University in Stephenville, Tx.

- 774. Les Davis
- i am an engineer who has had the usual series of calculus courses but not a course in analysis. i'm very interested in this subject and sincerely thank you for offering interactive real analysis. i shall return -les

- 773. rahman
- give a free book of real anlysis

- 772. Turan Yuksel
- Um, it seems that this pages are here for a long time, but I've found them today. Very nice, but it is not much generalized to n-dimensions (not hard work). Supplementary stuff may be added to make it a very complete book, a book of n-dimensions and including more of the basic information (such as topology, metric spaces). Can this book be mirrored, given the main site and creators' name and without any changes on the original document ?

- 771. rayda
- I am searching for a proof that show 2 to the power of n minus 1 equals the least number of moves it takes to move n blocks, stacked according to size larger on bottom and smaller as they go up, from one of 3 spaces to another..."Towers of Hanoi" If you can prove or can suggest link email me. Thanks R.

- 770. Curt Storlie
- I love this site. Finally a site where pure math students can frolic about

- 769. Michelle Poulos
- I am currently taking a video course in Real analysis. It is great to see that I can find information that will help me to understand the material. next semester I have to take vector analysis and I will definitly check out this site for help. Michelle

- 768. David Winsemius
- Regarding the rationals between 0 and 1 I believe the final statement lim sup (.)=0 should be lim sup inf (.) = 0.

**(Fixed - thanks; bgw)** - 767. David Winsemius
- 766. chris
- I enjoy your site but I had a similar question as another person earlier on. To avoid having to stay hooked up to the web is there any way to download the pages via a zip file, so one can peruse without the net?

- 765. Neal Seidl
- This site is wonderful!

- 764. Enrica Palmieri
- Veramente molto interessante, anche se ? difficile tradurre tutto in italiano. Sarebbe fantastico avere la versione italiana.

- 763. Lorenzo Sutton
- Very interesting and usefull

- 762. Alief Surachman
- This is one in very short list mathematics-related reference available on-line, that I really appreciate. Thank you very much of your hard work and kindness.

- 761. Melissa S Moreno
- I am a graduate student getting my masters in Mathematics. Your website has been extremely useful. I wish there were more like it. How about creating a Linear Algebra site?

- 760. Matt Mason
- Cool. I did an internet search on "Lipschitz continuity" and it brought me here. Thanks!

- 759. Ingo Dahn
- It is a very nice site. However, I could not get the applets displayed within a reasonable time. I am working on a book on calculus (in German) with Internet support. (see http://www.itaw.hu-berlin.de/sbt.htm) I should like to include some links to your site. This would assume that the structure of your site is rather stable. Please, let me know, whether this is the case. With compliments, Ingo Dahn

- 758. Justin
- I love this site this site rules. It is educationaly fun and interacting

- 757. kevin khan
- great job, can't wait to see the finished project with metric spaces. I hope it will be as informative as all that I have read so far.

- 756. DouglasHoward
- I have found your site very useful in gaining understanding for my Advanced Analysis course here at TSU (Tarleton Stater University). Thank You for making this site availiable to everyone.

- 755. Danielle King
- my professor sucks and this site is a life-saver!

- 754. Heather Torres
- none yet, just visited for first time - will write more later..

- 753. Ivan Soto
- I hope you can keep this inititaive going ... Best wishes.

- 752. Scott Garinger
- Mind blowing

- 751. Mike Brown
- This is an absolutely wonderful site. For me, it allows a basic review of portions of Real Analysis I had long forgotte. Thank you. Dr. Michael L. Brown

- 750. Juha Santala
- It`s great ,that somebody makes this sort of sites!This really helps studying.

- 749. Alan Polis
- 748. Julie Bohm
- 747. Marlet Castillo
- 746. Christopher Chan
- Good Work!

- 745. Joseph M. Hobbs
- 744. Phil Sayre
- I am pleased to find this site in InternetLand. I am retired (age 70), still interested in advancing my knowledge of mathematics. Especially liked the history tidbits. Will you be doing a chapter on special functions? I'll be back.

- 743. Cherie Courseault
- I like the site. I wish it had more proofs.

- 742. Bill
- attempting to learn more math for its aesthetic value, this site is of good quality

- 741. Kristin Koetz Carp
- I was searching for a solution for presentation format. You provided a methodology that was foundational to my program. I humbly respect your desire to provide information easily and.

- 740. Pi
- Great web-site. I was looking for history about great mathematicians and you've got it going on! Thanks, Pi (6th Grader, North Carolina USA)

- 739. Eric Wikner
- Democratization of education through presentation of quality courseware like this is profoundly important for world societies. The entrance threshold for educational opportunity is slowly moving from "incomprehensible affluence" to "access to a public library". Thanks.

- 738. Scott Martin
- Thankyou so much for this on line text. In short, it got me through my real analysis class. Once again, thankyou.

- 737. Robert Seffrin
- 736. Benito J. Gonzalez
- 735. Ian Hu
- I have a linear algebra assignment due morrow. The section on sets, injection, surjection, bijection, is terrific. Still confused about equivalence relations -- for example, for reflexive, what exactly does it mean that every element with a relation has a relation to itself? solid examples would've helped for beginners...

- 734. RomanX
- 733. Maurice Corlandi
- 732. Jack Tseng
- You have made it simple, additional pictures will help! This is a great way to learn on the web! Thanks for the material.

- 731. jon morrison
- 730. Larry Cohen
- Outstanding site. Very imaginative way to present mathematical concepts.

- 729. Hans Roos
- I'm a math teacher from the Netherlands, looking for some facts about Taylor's Theorem

- 728. Reid Guenther
- Thank you very much for writing (I use this term loosely as what has been done here is much more than a typical textbook.) this interactive text. I am afraid that after I go through this online 'book' that I will find reading normal text books to be a very tedious task. :-)

- 727. Raphael Goh
- Thank you for ur precious information!

- 726. Fabio Maria Manenti
- 725. Markus Wahrheit
- just let me thank you for this great site. awesome layout.

- 724. bj?rn harstad
- 723. Javier D. Burgos
- an excellent effort, congratulations!

- 722. ERCAN GURVIT
- It's a very carefully organised site. But because of the characteristics of our education system, I have to reorganize it according to our education program and add many numerical examples. Anyway It's a very helpful source for everybody concerned with the subject.

- 721. elidrissi
- An excellent and very Interesting site.

- 720. Susan Barocas
- Looking forward trying this out!!

- 719. Raj
- As a maths graduate myself, I find this site very informative. An addition would be to cover simpler topics such as differentiation an integration, curve sketching. This would allow more people to visit your site. Thank you.

- 718. Ed Romana
- Thank you very much for this accessible Encyclopedia of Real Analysis. It is a pleasure to see historical notes included which add a useful perspective to the subject. This web site is definitely on my browser's bookmarks. As a modern tool to practice mathematics maybe a section could be devoted to such software as: MACSYMA (my personal preference), MATHEMATICA, and MAPLE. I hope that the sections "under construction" will be completed, in time, with the same brief clarity that is evident thoroughout. Best Regards, Ed

- 717. Niek Sprakel
- Great site!

- 716. Marcel van Velzen
- 715. Jason Foulkes
- 714. Renato Alvarez-Nodarse
- A very interesting pages!!!!

- 713. mike alder
- Nice Layout. Hope you get my email.

- 712. Mustafa Kalafat
- It's perfect, you've done a very nice thing. Something like this may be written in another areas.

- 711. Rick Lichtenfels
- Interesting site...very informative.

- 710. Luis A. Armendariz
- Excellent job. Having an online textbook is an excellent idea, especially considering the mathematical content. The applets are awesome. They'd would also be really useful for visualizing concepts in the analogous field of complex analysis. Any ideas? I'm also curious to know what algorithm you've used for parsing the function (user input) I'd appreciate any help since I'm also working on a similar site (in content) and I'm writing my own applets for that purpose...

- 709. Jaime
- 708. Margaret M. Francis
- Superb concept, nicely executed. I appreciate the lucid, coherent approach (as opposed to the more common approach :-) in this area. I will refer students to your site as I believe it will be useful in learning Discrete Math and Theory of Computation. BTW, ending up at your site was pure serendipity. Seeking some mindless entertainment, I tried my internet provider's "surf according to your mood" toy...I followed a couple of links and ended up at your site :-) Not mindless but a pleasure to read nevertheless.

- 707. V. Simmons
- New visitor.

- 706. Atique Malik
- I am an engineer , and I was searching the Web trying to find the definition of a compact set. Thats how I got here. I like it!! Good work!!

- 705. Johan Mattsson
- These kinds of sites really makes you feel that that there really is something worthwhile on the net ( after all ... )

- 704. Andrew Knyazev
- 703. Bill Adams
- This seems to be a great idea. I went through the first couple of chapters. The ideas in the infinity chapter are very well presented. Nice work and Thanks. I'll be back

- 702. Anthony Will
- I will send comments as I have the opportunity to evaluate this tutorial. It looked interesting, so therefore I am signing in. Thank you.

- 701. lorena lim
- 700. Jeff Rudin
- 699. Dawn Goldson
- I don't know about anyone else out there, but advanced calculus is one thing I would rather do without. However, this program has been a literal life-saver!!!! Please keep it going!!!

- 698. Mikael DK
- 697. Luca Brandolini
- The idea of writing a WWW version of a Real Analisys book is very nice. The book also seems very nice L. Brandolini

- 696. Eduardo Rocha Jr
- I found all your information very useful in writing my research. Thank you for you hard work. It is really appreciated by us calculus students.

- 695. Irfan Ansari
- 694. Tom Reuterdahl
- I've just found the site and have not looked at the materials closely yet ... I'll send comments later. Tom Reuterdahl, Professor of Mathematics, Skyline College

- 693. Wilfried de Waele
- 692. oyd 11
- gr8 site, really help'd me nearly 2 years(!) ago for my 1st analysis in my 1st math year, but hey, i thought that the missing texts'll be there in MAX 5 or 6 month... y r u guys taking so long w/ it?

- 691. jason zegestowsky
- 690. Tom Klein
- I was very surprised to find a site containing such a user friendly and seemingly (to me, now) complete 'text' on both differential and integral real analysis. I've bookmarked your site and plan on coming back to give it another try.

- 689. Dennis Smoot
- Dear Persons, This is truly a great work. I am a graduate student in Mathematics and now that I know of your site I will read it and use it to review. I think there is a great need for readilly accessible knowledge like this. Thank you.

- 688. William Gempel
- Thank you for all the work you put into this. Is there a forum for questions?

- 687. David Harvey
- 686. Eric Roach
- Nice site. I wish I could find analogous ones for physics. I got here via link from 'The Continuum Hypothesis' by Nancy McGough, www.best.com/~ii/ch#overview

- 685. wayne
- 684. dee
- 683. Hon Shiu Costello
- 682. Ronald Monson
- Quite impressive: The Java tools didn't seem to work smoothly; I'm using Netscape 3 on a Mac 68K. More Detailed exercises and answers would be useful.

- 681. Rachel Bowman
- 680. Giedre
- 679. Alexey Kosenkov, MSTU
- Hi, Iam is great mathematical fan!! I LOVE this science and very pleasant to any letters from same fans, that may want send it to me!

- 678. Pedro Meza
- i am a student, and liked your place very much! Nice work, also thanks, bye

- 677. lmanuel
- I am very impressed with the information provided. This information worked as a supplement for my applied analysis text. Once again many thanks!!!

- 676. Petri Ojanen
- 675. Samantha Feakins
- 674. Richard Bournes
- Thanks for an excellent resource. As a mature-age undergraduate, I am impressed with the clarity and conciseness of the material. I arrived at this site because I was searching for material of sequences and series to supplement my lecture notes. Congratulations on an excellent presentation.

- 673. Scott Horn
- I like this site. I don't have much time play with it because I am a teacher. . . You, understand.

- 672. Rolland Thill
- Greetings from Herscher Illinois

- 671. Laura Thompson
- Nice page. I think there is too much pointing and clicking. Maybe the classic thms don't need the "hidden" proof. It's almost as though you expect the average student to attempt proofs of thms like Bolanzo-Weierstrauss. Of course, it might be beneficial for them to think about the thm first instead of immediately going to the proof, but hopefully, they've seen it already in class. I still prefer pencil, paper, and (several) real textbooks, but would certainly use this page to locate interesting techniques of proof....just as I might do with any new textbook.

- 670. Harvey H. Green
- I've just begun so my comments wouldn't have much value now. You may be interested to know I'm a 63 year old retiree who doesn't have much recall of his now ancient (and very un-modern) Advanced Calculus course.

- 669. Dr. Sergey Belui
- 668. V. S. Somayazulu
- 667. Roldano De Persio
- 666. mark hunt
- 665. Adenone Diniz Costa
- 664. Christopher A. Barker
- Visiting your site as an external reviewer for your CDI grant proposal--specifically your Java module on Newton's method.

- 663. Dr. David R. Bickel
- I like the glossary.

- 662. Reggie Farmer
- I need REAL help

- 661. Denise Duffy
- A terrific site!! I have been able to follow the course without ever having a formal real analysis class.

- 660. Antti-Juhani Kaijanaho
- I haven't read the text, but it seems like good work. Thanks. However, I would like if the usage of the symbol '#' (apparently as the symbol for nonequality) would be clearly stated.

- 659. Daniel Trambaiolo
- The site looks great, and I look forward to seeing more of it completed. It might be a good idea to put at the top of each chapter a list of prerequisites for the understanding of the chapter, since for an online "text", it is unlikely that all people will read it in a linear way.

- 658. Fan Tou
- 657. Dhvnaika Gandhi
- Why is a geometric series called a geometric series? (What makes a series geometric?)

- 656. kendra riddle
- 655. Eddie White
- The 'book' looks great.

- 654. Jouni Savolainen
- 653. Steve Burke
- 652. Jim Kaus
- 651. Teemu Nuutinen
- This site is the best math archive I've encountered. It has maths displayed "as it is" and not as messy ascii graphics. I found this site accidentally while trying to search information on Weierstrass' function. An outstanding found! I'll be visiting this site a lot more from now on.

- 650. Karin Davila
- 649. Andy Lewis
- 648. Ross Ritterman
- Being an AP Calculus BC student, its cool to see these resources on the internet. Good Stuff, well done.

- 647. Daniel White
- I am just starting Honors Analysis, and this look slike a good place to go to for reference or additional problems.

- 646. israel Peralta
- Great site full of usefull info. Thanks

- 645. Salbando Jones
- Excellent! I will recommend my AP Calculus students visit it often. Thank you.

- 644. Mike Coco
- 643. Carlton D. Brown
- Great site! I had a lot of difficulty seeing equations and concepts, but your graphs and illustrations are very helpful. Thanks

- 642. Donald F. Ryniker Jr.
- 641. Peter J Hazlett
- My professor is using IRA for our Honors Calculus course at Bridgewater State College, and so far I find it quite interesting. Keep up the good work!

- 640. herman knoop
- 639. michael burke
- 638. Alex Telford
- Excellent ! This is getting towards the 'Network Encyclopedia of All Mathematics' (which is what we want). Thank you for a very professional site.

- 637. Joe Mesko
- When will the book be available? I would love to purchase a copy. Please drop me an email when it is available. Thank you. Joe Mesko

- 636. Frank B. Fezza
- The Real Analysis course is great! I now understand concepts that were problematical for me in the past. I see a great deal of work... and thought... went into the creation of this course. Here is one math major who has a better understanding of topics in Real Analysis. Thanks again for your contribution to learning!

- 635. Parth
- 634. Edgar Lindamood
- I'm a 48 year old manager with a M.S. from Va. Tech in math. My concentration was in Topology and I decided to try my hand at brushing back up on some Reals. This a great intro for getting back into the material: Great stuff!

- 633. Kate Shaw
- 632. Kyle Moore
- 631. Peter D. Doucette
- A great resource!

- 630. Karen Lerche
- 629. Thang Doan
- 628. wallop chalerm
- 627. Matt Ziegler
- Great Page!!!! Like the Java Tools

- 626. Ross Ritterman
- I'm an AP Calc BC student and the stuff here is great as a supplement!!! Well done. I'm especially fond of the function plotter applet.

- 625. Rich Reeves
- Cool!

- 624. Suda Kunyosying
- 623. Harvey Greenberg
- Excellent contribution!

- 622. Keith Leatham
- 621. Dorothy Takahashi
- This is great. Thanks for the info.

- 620. Jamie Wallace
- Very interesting

- 619. Karen Saxe
- 618. Carmelo Ruiz
- Thanks for your excelent work. I wish there were another interactive-books about other subjects in Maths. Thank very much

- 617. kara lloyd
- O.K. Can you please explain why all math books, and this sit avoid doing proofs?? They are not always so obvious. Why are they always left to the reader. And how do you prove multiplication axioms in the set of all reals???? Otherwise cool site.

- 616. Melanie Hancock
- I am in Analysis right now, and your sight is really interesting. But we need to prove that multiplication axioms hold in the real number system. We came here for advice on finding the correct cut to use as an inverse. ANy help would be greatly appreciatted.

- 615. Daniel A. Deadwyler
- 614. Mark Tillinghast
- Maybe the limit of an alternating real series is complex: Lim (-1)**n = i n-infinity ? A definition is not necessarily the truth.

- 613. Michael A. Saum
- This is a REALly excellent educational tool!

- 612. Terry Sargent
- 611. Revathi Narasimhan
- Nice format from which to learn the essence of advanced calculus. I shall direct my students to your site! Reva Narasimhan, Math, Saint Peter's College

- 610. Shawn P. Philippon
- Nice work. Well organized and written well. I would like to see more problems in each section, but I suppose I could go to a text. Thanks

- 609. Dennis D. Duquette
- I have visited the site www.shu.edu/projects/reals and have found it interesting. I think it is well organized, with the road map being a good guide for the user to see visually how the various topics are related to one another. The topic that fi rst got my attention was in the frame with the topic links, with the heading "Metric Spaces," as I had at least some familiarity with the other topics but had never heard of this one. It was a disappointment to find out that the chapter is not yet finish ed. There is also no link to a definition of metric spaces in the glossary. I think I shall be visiting this site more,as time permits. This first visit was an assignment from Professor Moreno to my Advanced Calculus with Applications class, with the abo ve being the paragraphs I wrote for him. I hope the feed-back is useful here as well.

- 608. Laura Bates
- 607. James Lamkin
- 606. Michael C. Doherty
- Needed info on set theory for a seventh grade math class which the daughter of my child care provider is taking.

- 605. Alex Fainshtein, PhD.
- Unexpectably impressive!

- 604. Rizwan Assad
- 603. Peter Macedo
- 602. Carlton D. Brown
- 601. John A. Lynch
- Great help in prep for a test CalcIII (tomorrow). I have it bookmarked for frequent visitation. I will return.

- 600. Robert Burke
- I'm at Queen's University in Canada, doing some third-year Complex Analysis. I looked up "Convergence Root Test" and found your site, and I'm blown away. I don't have to run home and grab my text now - your treatment of Real Analysis is more comp lete! Cheers and thanks.

- 599. James Daniel
- Calculus is my life. I worship it. I am a dork. Please send me any information or new theorems to keep me updated regularly.

- 598. Tim Manning
- Excellent resource - very helpful.

- 597. jeffrey robert johnson
- 596. serge de Gosson
- Fantastic site ! One of the best I have seen. This is what internet really is about and not for totally useless info. Are there other mathsites of that kind ? I would recommend them to my students. Keep up the good work!

- 595. Dr.T.N.Shanmugam
- I am yet to visit your pages as I have seen the comments of other I shall let you know my comments after browsing through your pages

- 594. David Weinbaum
- Congratulations! I'm a PhD candidate in Mathematical Finance at NYU, and I find you've done an excellent job of introducing Real Analysis in a rigorous yet readable fashion. I'm looking forward to reading your chapter on integrals. Also, are you go ing to treat Hilbert Spaces, the Riesz Representation theorem and Randon-Nikodym derivative? derivative

- 593. Stan Zietz
- 592. Vania
- Hello Bert, I am an assistant prof. at UT-Austin and since I learned about your site I never fail to mention it to my "Intro. to Real Analysis" students. So, I take the chance to congratulate for the great job you did! And here's the little typo I' ve found: Archilles, the guy who races against the turtle, should be spelled Achilles, without 'r' :). Thanks again, and greetings from a fellow European, Vania.

- 591. Janet Russell
- The information provided here is helping immensely with my courses this semester! (I wish I had found this site when I was taking discrete math last semester! Thanks for the help!

- 590. Giovanni Barbi
- Wonderful!

- 589. Renata Sarno, Ph.D.
- 588. Peter Finderup Lund
- I like the idea...will try it out in the next couple of weeks to see if it works:) (note the e-mail address will change to firefly@diku.dk after November 1 1997) -Peter

- 587. Klaus - Peter Banze
- 586. James Raskowski
- This is great!! I may have majored in math had I found this sooner....:) J.R.

- 585. Elton Pedersen
- 584. Al DeGennaro
- I am particularly interested in discrete mathematics, deductive logic, multi-valued logic, and all aspects of computing.

- 583. Dr. Paul Fairbanks
- 582. Moti Levy
- 581. Santiago Pinto
- When do you think will the other sections be finished?

- 580. Armand Martin
- 579. Dries D'hooghe
- Superb, it saves me from having to find my Math books

- 578. Geoff Taylor
- 577. Stephen B. Robinson
- 576. helix
- This is an excellent site. One of the most useful paradigms I've seen

- 575. C.P.
- You probably should state that division by zero is not allowed, in your definition of the set of rational numbers. C.P.

- 574. Dr. Hossein Arsham
- Your site made my teaching much easier. Thanks for your efforts.

- 573. Santiago M. Pinto
- Excellent job. I am very anxious waiting for the unfinished chapters. Excellent the interaction with the Java tools. Maybe you can include that in some other sections. Have you been thinking in the possibility of downloading?

- 572. K.Anderson
- 571. Fan Tou
- Very interesting and valuable materials. I am studying the material on my spare time as my self-improvement pragram. Thank you very much. Problem: I am having difficulty in printing the text, but I could print the examples and proofs.

- 570. Daniel Garrick
- 569. purvin vakharwala
- I really like the knowledge you have and i hope to gain some of that from you. I am in your class monday wednesday and fridays in lab at 2pm in INTRO TO COMPUTER SCIENCE I . Purvin Vakharwala

- 568. Russ Goodman
- 567. john morvay
- Since I do not use the word "cool" I will just say - outstanding. I will be coming to this site often.

- 566. ben kafka
- 565. Carlos Pacheco
- I wish there were more interactive boooks like this one. They are a goldmine for those who seek independent learning, or simply a taste of what a subject in mathematics is like. Keep up the good work! Carlos Pacheco, Mathematics, Univ. of Texas at Austin

- 564. ABM
- what a splendid site! good educational stuff!

- 563. Lawrence Roseman
- Excellent material from what I've seen so far. I plan to be back again. By the way, my Mom is a graduate of Seton Hall !

- 562. James Spencer
- 561. Bob Woodle
- VERY Impressive!

- 560. Michael Hyams
- Excellent interactive education!

- 559. Michelangelo Grigni
- I find this project very impressive, I hope you finish it. I am not sure if the web is perfect suited for replacing traditional textbooks, but I am very impressed with the potential of the interactive parts. In particular, is the source for your J ava tools available? I am interested in developing similar tools for some local courses.

- 558. Cynthia Garde
- This is utterly amazing! I never thought that such knowledge could exist at my fingertips. Thank God for Technology and great advancements; I am thankful that such grants, seton, were given to fund a gift to those who thirst for knowledge, espec ially within their own fields. Keep it up! Cynthia

- 557. Scott E Brodie
- Great Site.

- 556. Waqar Malik
- Hi Dr. Wachsmuth, I was reading some notes on your home page, and I have a question regarding uncountable sets. I apologize for disturbing you. Let P(S) be a set of permutations on S, ie. P(S) is the set of bijections S-S. Prove the P(N) is uncoun table. N is set of natural numbers. How would you go about it, and what would be the proof? Thanks, Waqar

- 555. Ong Ming Yuan
- Oops. I wasn't reading things carefully. You did state that c is in the closure of the domain. Do ignore the previous comments. Sorry.

- 554. Ong Ming Yuan
- There did not seem to be any thing about when the definition for limits is meaningfull. ie, that a limit at a point is only defined at a cluster point ( or a limit point ) of the functions domain. Some introduction should be provided as to such thi ngs. Also I think it would be better if, in the proofs of limit theorems, like lim (f(x) + g(x)) = lim f(x) + lim g(x) exactly on what domains are the functions defined. In fact domain ( f + g ) = domain (f) intersect domain (g) and so on. All in all, I think attention should be given to when definitions are meaningful. Anyway, it was fun down at this place.

**(Please see Ong Ming Yuan's other comment above; bgw)** - 553. Brian Carter
- THANK YOU! I am currently struggling through MTH 253. This web site has really helped. Thanks again, Brian

- 552. Marcus Henry
- 551. David A. Tomb
- A delightful and useful gift to the field.

- 550. max condiotti
- 549. Dennis Coffey
- 548. Donald McCracken
- 547. Paula Lopez
- I haven't been to this site before so I don't realy have any comments. But I'll let you know when I do.

- 546. Mark Whitacre
- Thank you for such a speedy response. The information will be very helpful. I have a small physics web site in progress, but I'm anxious to get started on my analysis site. Thanks again,

- 545. Mark Whitacre
- Dr.Wachsmuth, I really like your real analysis web site. I am trying to create web sites for my mathematics and physics classes. I am wondering what program you use to create your math symbols. I am very frustrated in this part of the process. Any help would be greatly appreciated. Thanks, Mark Whitacre, Physics & Analysis Instructor, Lebanon High School, http://www.llion/~mwhitacr/phy/physics.html

- 544. Steve Leffler
- I'm a scientist working in advanced construction technology. I just found your web site, and I think it can help find solutions to practical application problems. I will add this as a bookmark and explore it in depth later. Thanks, Steve

- 543. jhusein
- In the exercise on relations g(x)=arctan(x) the answer given was that arctan(x) was one to one. Doesn't arctan(x) actually have more than one value, eg: arctan(1)=45 deg or (45+180) deg . Is arctan(x) a function if you define the range as R?

- 542. Dan Hahs
- Excellent presentation.

- 541. Doug McKenzie
- This is the coolest thing on line. I have been studying some math on my own, not having the time to attend classes, and really wanting to do it at my own pace. This is great! I will be back... thanks!

- 540. eee pacheco
- Interesting site, but would like to see more of the humanistic side of the persons, mathematicians

- 539. Dorothy Takahashi
- This is great. Thanks for the info.

- 538. Amir Shimoni
- Dear Prof. Wachsmuth, I was looking at the "Interactive Real Analysis" and I noticed something (I think is) wrong with Euclid's Theorem (There is no largest prime number. ) You say: "and therefore M is not divisible by any number. That means t hat M is also a prime number...". This is false... because M is not divisible by any prime from 2..N doesn't imply that M is prime. It might be divisible by a number between (N+1)...(M-1). This still proves that N is not the biggest prime, since there is either a prime in the range (N+1) ...(M-1) that divides M or M is a prime. I've seen this proof written so many times like this, and just bugs me sometimes... Pet peeve, I guess, Amir Shimoni

**(After some discussions, we clarified the problem - the proof that's in the text was and is correct, and illustrates "proof by contradiction", which often leads to confusions; bgw)** - 537. Pierre Bdcklund
- 536. Robert Lance
- hot links on the road map would be useful

- 535. Simon Clippingdale
- Admirably clear presentation. Keep up the excellent work. Cheers.

- 534. Bill Pappan
- 533. Brian Brake
- Love it! I'll let everyone know at Cal Poly's Math Dept. in on my little secret. Perhaps this site might be improve by making it more interactive. For example, add at the end of each topic page a self test for review. This will ultimately challe nge the students to retain & use all the theorems provided.

- 532. Nathan H. Brown, Jr.
- This is an exemplary site. It provides an excellent introduction to a diversity of topics for the undergraduate math student and others working in disciplines dependent on a clear understanding of certain topics in real analysis. I have found it to be a great help in the clarification of definitions in topology. The site also provides clear examples and proofs of key concepts in all supported areas. Keep up the good work.

- 531. Jay C. Caminero
- Your site is really great.... Lots of stuff that I need, though a friend of mine got to search and found out your site first (before me). I'm taking Advance Calculus... and many of the topics discussed are IN here...If not I, we will surely be ba ck here and check out how you prove them. More Power!!!

- 530. Cem Say
- Dear Dr. Wachsmuth, Thanks and congratulations for the real analysis book. This is what the Internet was invented for. Great work. Best, A. C. Cem Say, Associate Professor, BoGaziCi University, Department of Computer Engineering, Bebek 80815, Ista nbul, Turkey

- 529. Jean-Christophe Nave
- 528. Joseph Merton
- 527. Luke Mawbey
- This site is a life saver! Finally, a resource which will allow me to learn all of the work that I should have done in class!

- 526. steven a miller
- I would like to see the excercises worked out in other words the proofs formally written up.

- 525. Chris Quartetti
- Nice resource... I found it looking for a good definition of "supremum" in HotBot. I'll definitely keep this bookmarked as a reference.

- 524. Mark Jensen
- Excellent site -- I especially like the historical tidbits. One criticism is the communication rate seems to be very slow -- is this on your end? Other than that, I'm very impressed. Keep up the good work!

- 523. Mark Spencer
- I was excited to find this site. I plan to use this interactive textbook along with my text this fall when I take Real Analysis I at UTSA. I'm counting on this to increase my understanding of the subject.

- 522. Charles T. Mosier
- 521. jodi frietag
- 520. Karin N Davila
- 519. Mark Wittig
- 518. Don Oppliger
- Stumbled onto your site while doing a search for information on Goedel, so I stayed and browsed a bit. Wish we had had this sort of thing available when I took my degree almost 30 years ago...Nice to see that math remains the same!

- 517. Sin Boon Wah
- 516. Alfred Barron
- I took advanced calculus in '83 using the book by Rudin. My favorite backups were by Spivak, Chinn & Steenrod, and Marsden. After going through some of your text I can only wish that it was around then. This indeed is a gift to all students of math ematics throughout the world. Informal in style, but sound in content. I look forward to your section on the Lebesgue integral. My only recommendation is for _more_ problems and solutions, and references for each topic. Hey, where's the probability ? :-)

- 515. Krishnendu Dasgupta
- Excellent online resource. Wish I had such a dynamic, comprehensive reference while I had Analysis in college!

- 514. Derek Prowse
- Brilliant; thank you.

- 513. Matthew S. Westerhoff
- Your online notes are good.

- 512. Steve Shore
- Nice job, especially the historical discussions.

- 511. Herman van Dorsten
- 510. Greg Payne
- "Interactive Real Analysis" may prove a lifesaving tool for my Real II exam in six days. Something on critical point classification using the Hessian (2 variables) would have been nice. If not my life, it's certainly saved me $A69.95 for the tex tbook. A most dedicated effort which deserves my commendation.

- 509. Jay A. Kaplan
- Wow, this is cool !!

- 508. Howard Blankfeld
- 507. Gulden Kokturk
- Very well html side. Information about real analysis is clear and understandable.

- 506. Jose E. Burgos
- B.A. Psychology, Universidad Catolica Andres Bello, Caracas, Venezuela (1983). M.S. Experimental Analysis of Behavior, Universidad Central de Venezuela (1989), Ph.D. Neuroscience and Behavior, University of Massachusetts (1996). Currently working o n neural networks and genetic algorithms (basically combining them to simulate the evolution of Pavlovian conditioning). Also *very* interested in abstract mathematics (formal logic and set theory) as a tool for systematic philosophical discussion.

- 505. Helene Karin Buset
- 504. Abe Mantell
- NICE SITE!!!

- 503. Brennan Glen Sebastain
- I've just got to this site. It looks very promising. I hope it is.

- 502. Chris Tweddle
- I am currently researching a paper on the rigorous development of real numbers. However, I am finding it difficult to find a thorough presentation of Weierstrass's theory of irrational numbers. If you know of such a source (in either English or Fr ench), I would be grateful for the information. Thank you for your time, Chris

- 501. Chris Rao
- 500. Stas Klimenko
- It is excellent site for students and for everybody who need in recalling a math background. Thank you very much. With the great esteem, Stas

- 499. SuZanne Freyder
- This is a wonderfull concept. I am taking an independent study durring the Summer and Fall semesters and this website has been/is a great resource. I don't have to search thousands of thousands (ok make that more like hundreds of hundreds) of boo ks just to find something on a topic. Thanks for your help in my studies.

- 498. Ranjan A. Pillay
- Very, very, impressive web site. To all who contributed to this magnificent web site, hello. My name is Ranjan A. Pillay. I am a calculus student at Cloverdale High School in California, and I somehow found this page. As I said before, this page is truly incredible. I was searching for a little bit of help, if not an answer for a certain problem in my Calculus class. We have been touching on derivatives again, and we were given the following problem: If y = X^X [X raised to the X power] then fi nd the derivative function. What is y' = ? I tried every imaginable method to approach this problem. I treated it as a power function and an exponential fuction. I currently have not a clue. Please, do not feel obligated to answer this question. I believe it is my extreme "end-of-the-year" stre ss syndrome that caused me to email this. Anyway, I will continue to wallow over this one. I still haven't gone through all of the web site. From what I can see know, I should be able to find the answer to this specific problem. Thank you for your time in reading this. Cordially, Ranjan A. Pillay. P.S. Did I mention that this was a very impressive site?

- 497. Bryan Seale
- Handy info, thanks!

- 496. Theo Calis
- I'm very curious

- 495. COLOMBO
- 494. Otto Rubingh
- Looks great !! and can be very helpfull

- 493. Jennifer Wooton
- Very helpful!!

- 492. Bill Lagarde
- 491. Rachel Montague
- This is very well organised, and useful for people doing reports like me. It would be easier if you could get to any site from the road map.

- 490. David Tomb
- This may be my favorite site

- 489. Claudiu Remsing
- 488. Wesley Elias Ribeiro
- Your Interactive Real Analysis is just great ! Please consider adding other subjects too (not only math, but also physics, chemistry, ...).

- 487. Llums Arilla
- 486. T. Michael Rhodes
- I think that this site could serve as a model for instructional and/or enrichment web sites for university level mathematics. I would like to see some of the topics included in your development covered in more detail, such as, greater nos. of exam ples, more thorough explanations, etc. But in general, your site is outstanding !

- 485. Feng Gao
- 484. Erin Pearse
- I'm currently taking a first-year course in Analysis and we're going through Rudin's Principles of Mathematical Analysis. Could you recommend a good web site to visit for information pertaining to chaptes 9 & 10: Vector Spaces and Differential Form s? -Thanks, Erin Pearse

- 483. Jeff Anspach
- 482. A Raudzens
- Please proofread your website!

- 481. Eileen Kraemer
- Very nice glossary for quick reference.

- 480. Nao Wakamatsu
- This is a great web site.

- 479. maura lusk
- 478. Nathan Pochucha
- Interesting web site! I enjoyed surfing your web site dudes!

- 477. Donnell Kwong
- Text-content is extensive. Needed more interactive animated visual presentations. Thank you

- 476. giacinto orlando
- 475. Edward Chiang
- 474. Dan Snyder
- 473. Vmctor M. HernBndez L.
- Estimado Dr. Wachsmuth, El encuento con su Interactive Real Analysis results para mi una muestra de la armonia que puede lograrse entre las matemBticas, la tecnologma y un prmstino sentido estitico. Su trabajo me ha representado un estmmulo para seguir cultivando las matemBticas, la tecnologma y mi sentido estitico. Felicitaciones desde Hermosillo, Sonora. Mixico.

- 472. Raina Johnson
- This is one of the most interesting websights I have seen on math. Your use of examples is wonderful. I wish I would have found this websight at the beginning of this semester.

- 471. u8308048
- i come from taiwan ur page is good to a student

- 470. Alon Barlevy
- Dear Prof. Wachsmuth, your Interactive Real Analysis Site is pretty informative. I have one question though on the function f(x) = x sin(1/x). In your section on continuity, you ask what type of discontinuity does this function have at x=0. The ans wer you give is a removeable discontinuity. I never took real analysis, but I have taken complex analysis, where we studied the Laurent Series. The Laurent Series of this function [f(x) = x sin(1/x)] has infinite number of terms, hence the singularity sh ould be essential. For details, I reffer you to the text book we used in the Complex Analysis class. The book is R. V. Churchill and J. W. Brown, "Complex Variables and Applications", Fifth Edition, McGraw Hill, 1990, page 173. Am I not understanding some thing? Thanks for your help, Alon Barlevy.

**(Good question. However, there are important differences between removable discontinuity and essential and other singularities as they appear in a text on Complex Analysis; bgw)** - 469. Alon Barlevy
- Your book (site) is pretty cool. Please let me know when you get the pointwise convergence page set up. Thanks.

- 468. Naveen Mathew
- The book was very informative. I found a lot of interesting stuff on the topic i was searching for. I'mm sure I will come back here when I am looking for more neat stuff. Thanks a lot

- 467. Sean Rice
- Thank you very much for these web pages. I have a few books on the subject of Real Analysis but they are no where as complete as these pages. I am now on a personal quest to study the theory of wavelets and have forgotten the exact meaning of many terms in real analysis. The ease of walking through these pages greatly helps in remembering and learning these ideas. Once again thank you for creating these pages. I am looking forward to when you get the area of "Metric Spaces" done, as General Relativity and Tensor analysis is another area I am on a personal quest to learn.

- 466. Lars Lundvall
- I welcome Your initiative to present this difficult subject on Internet. Sincerely Yours, Lars Lundvall, Stockholm

- 465. Petra Hagan
- 464. SuZanne Freyder
- I wish I had known about this site at the beginning of the semester. This sight is great!!

- 463. Ricardo
- 462. Mike Herndon
- 461. Dan Gold
- Very cool site!!! Impressive work! Thanks.

- 460. Brad Martin
- Wonderful site. I wish I had found this earlier in the semester. Thanks.

- 459. Jaime Moreno
- Most useful use of Internet seen to date! An example to all disbelievers that question usefullness of Net.

- 458. Eric Kazee
- This is pretty cool. Useful too. Umm...not much else to say. :)

- 457. Nick Nielsen
- This is the kind of resource that makes the net a wonderful place. I'd like to see more on basic concepts of topology like the Jordan Curve theorem, etc.

- 456. Nicholas James
- WOW! This is the slickest mathematical website I've ever seen! I've just discovered it, so I have no constructive comments or criticisms yet. I'm really impressed!

- 455. Amity Warren
- Hello, I am currently a student at the University of California at Santa Cruz, and by way of looking for information on Real Analysis, found your wonderful interactive text book on the subject. I just wanted to say how much your book has helped me these past two weeks during the introductory part of my Real Analysis class. Please keep working on this! It is just fantastic.

- 454. satish raghavendran
- It is an excellent site for students

- 453. Math 453 Real Analysis at Hood College
- 452. Ju-Zhou Tao
- Dear Dr. Wachsmuth, Thank you very much for bringing such a chance of free learning to people loving mathematics. When are the sections after "The Intergral" coming out? I am looking forward to it. Regards, Tao (4/20/97)

- 451. Ulysses Sodre
- Excelent. Can I translate this page to Portuguese (Brasil).

- 450. Marianne Donley
- This was great!

- 449. Mickey Di Siena
- I checked out your page on the Pointwise Convergence and was very disappointed to see that it was still under construction.

- 448. SK Lua
- Great stuff! It is really impressive and useful.

- 447. nuccia
- desidero notizie su "Special Function of Peano" .Thank

- 446. Eric Roach
- I was disappointed that the page(s) on Metric Spaces is not yet complete. I am looking for a resource that discusses Tensor Analysis in some detail (at a fairly simple level). My background includes Complex Analysis and Vector Calculus

- 445. Gary Soibel
- I am very happy to see this project. I would also be interseted in on-line or electronic technical books.

- 444. Gary Wise
- When you treat equivalence relations, be a little cautious about the idea of transitivity. Recall that a set is transitive iff any element of it is a subset of it. For example, think of ordinals. By the way, why not introduce ordinals also ?

- 443. Steven Gross
- Doing some random browsing, there's a little problem on the page defining the Cauchy product c(sub n) is missing.

- 442. Steven Gross
- I haven't read enough to have anything insightful to say yet, but it looks terrific. The black on green parts of the map are very hard to read on my screen.

- 441. Chu Chengnan
- 440. Robin Kuns
- 439. Gary Wise
- Who are you? I've just glimpsed at this. It was recommended to me by a student I taught last semester. He was in electrical engineering. He took my course in probability, and now he is taking real analysis as an elective. I think this idea of a web page is pretty good. The web has needed something like this. Want counterexamples? Have you seen my book, "Counterexamples in Probability and Real Analysis " ? Perhaps you could touch on probability some. You might amaze some people.

- 438. SANTIAGO
- 437. Farrah Fuller
- 436. satish Raghavendran
- 435. Matthew S. Westerhoff
- I am very pleased with this homepage.

- 434. Jennifer Matson
- 433. Claudia Catalan
- 432. Robbie Trower
- I was told to look at this site as I am researching the use of java in teaching/ learning methods. I have found it useful and interseting. Thankyou.

- 431. C. Howard
- I've only skimmed your tutorial - I'm adding a link to it on my math resources page

- 430. Preethi
- Looking for java Applications Could be a best resource for my learn and utilize Java

- 429. ralph saxton
- very nice work!

- 428. Stacy Elliott
- I am currently taking a Real Analysis course at Appalachian State University in Boone, NC. Anything you can do to help me through this course will be greatly appreciated. :)

- 427. John Glidewell
- Fantastic

- 426. Lopaka
- Take some time to look through it. Very nice.

- 425. Maddy Baker
- Great Idea! I'm a grad student; this site makes real analysis seem (almost) fun.

- 424. A.S. Vasudeva Murthy
- 423. Mauricio de Castro
- Dear Sir, I want to have a copy of your wonderful program. I will pay for it. Thanks. Olt?chio Mariano Carneiro, SRES 10, J, 26, 70645-100 Bras?lia, DF, Brasil

- 422. Oltßchio M. Carneiro
- 421. Rooman
- 420. J.S. Kincaid
- Highly informative. I plan on returning to this site often.

- 419. Ray Beaulieu
- WOW!

- 418. Dennis Murphy
- I am using this as a tutorial for my computer class, CS251. This Program is interesting and helpful. Please feel free to send additional information to my email address.

- 417. Christine Hallberg Stewart
- This is a tremendous resource. I wish I had access to something like this when I studied Real Analysis as a junior in college! Now as a graduate assistant I will recommend it to others who are currently studying the subject. It also served

- 416. Steve Lewis
- Looking really interesting, the way the web should be. I look forward to seeing your progress. A few more examples would make it really an interactive learning experience. As a teacher of the subject myself I am looking for such things.

- 415. c. sakkas
- Congratulations! The text is concise and instructive, written especially for people who do not take anything for granted. However, the word "obvious" should disappear from our vocabulary. Nothing is "obvious" in Mathematics. Please more historical references to better motivate. Thank you for you patience!

- 414. Marvin Edwards
- 413. k. m. chick
- 412. howard bondell
- I am really excited that I found this site. Not only is it amazingly thorough, it is extremely informative as well. I am currently a junior math major at Rutgers University who has just returned to school after a four year absence. I am studying to take the calculus and linear algebra portion of the actuarial exam and need a lot of help in reviewing calculus. I have found this site to be of assistance to me in my journey of rediscovering calculus. Thank you.

- 411. Trudy Taylor
- I love interactive real analysis except that you don't have Saltus functions yet. Keep Truckin', Trudy

- 410. chy-bong chung
- Nice to see it!!

- 409. Philip C. Almes
- Wonderful! I found your page while checking on an assignment for Peano's Axioms. Keep up the good work!

- 408. Luca Mazzucchelli
- First of all the best compliments for your beautiful interactive book. Second: I've read about Zeno's paradox in the chapter about Series and I've found that you refer about "...Archilles and the Tortoise..."? Are you sure that the correct spelling is "Archilles" and not "Achilles"?? Sorry for this trivial remark.

**(Correct - my spelling has been fixed. Thanks; bgw)**Best Regards, Luca Mazzucchelli, (Milano, Italy) - 407. Irv Kaplan
- wonderful concept, well executed.

- 406. Mike McLendon
- I enjoyed this Interactive textbook a lot. I found your site searching for info on the projective plane. You use it as an example of an equivalence realtion in the Sets & Relations chapter. If you don't mind, I'd like to include a link to your page on my personal page. Thanks! Mike McLendon ( http://www.math.uiowa.edu/~mmclendo)

- 405. Alexander Theel
- I like Interactive Real Analysis very and will use it more frequently while learning for exams and working out worksheets. It's great.

- 404. James Post
- 403. anonymous
- 402. Arne L°ftingsmo
- I have just learnt about sup and inf with their limits form your 'book'. It looks very promising. I will probably use it more since I am in the process of learning stochastic calculus.

- 401. Gary Karg
- I have until Tuesday to learn Chapters 8.1-8.10 Thomas & Finney (9th ed) about tests for convergence etc. I hope this net site will be more readable than the text is !!! Thankfully, Gary.

- 400. Arun Kumar
- This is an excellent book. I have tried to teach myself real analysis from Apostol, Rudin, and other standard texts, but I always lost their thread of argument somewhere. That hasn't happened here, not so far. And if I do get lost, I suppose I could send you more mail.

- 399. Kriang Lerdsuwanakij
- Great work! I like it's simple language. Illustration and counterexamples helps a lot.

- 398. c bennett
- thank you for the information. keep up the good work

- 397. Paul Brow
- Very helpful

- 396. Humberto Madrid
- Nice work! I wish it will be completed soon! I would like to have notices of advances or improvments.

- 395. Doug Hockenberry
- 394. Tammy Wittig
- 393. David Keil
- Thanks for making this available. I teach a course at Mass. Bay Community College in Computation.

- 392. Jonathan Dunn
- A good idea, done very well.

- 391. Kathryn Ward
- I was hoping to find the formula for n(AuBuCuD) unfortunatly I couldn't find it but I did find other interesting information. Thank you for a great site.

- 390. Sheila Schaetzke
- Great site! I will be back

- 389. Dale E Brown
- very interesting, anxious to see more, will be returning from time to time, tks

- 388. Kenneth Luce JR.
- 387. Elsa Newman
- Wow... I haven't had time to look carefully yet, but this looks great!

- 386. Brent Granthem
- Thanks so much for the help. Your examples and explainations seem to make more sense to a struggling student in discrete mathmatics.

- 385. Brown Marc
- Fantastic

- 384. Alfredo Lanzo
- This is Faaaaaannnnnntastic!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

- 383. Dr Paolo Remagnino
- Seems good !

- 382. Kevin Higham
- I am very excited to have stumbled onto this page. I am a first year sstudent and I am sruggling. I can see that this 'text book' is going to be invaluable in helping my understanding. Many thanks

- 381. Luca Mazzucchelli
- 380. Alireza Bab-Hadiashar
- Dear Dr. Wachsmbe, I was reading your web page about the Zeno's paradox and I saw this: Now we want to take the limit as n goes to infinity to find out when the distance between Archilles and the tortoise is zero. But that involves adding infinitely many numbers in the above expression for the time, and we don't know how to do that. However, if we define S n = 1 + 1 / 2 + 1 / 2 2 + 1 / 2 3 + ... + 1 / 2 n then, dividing by 2 and subtracting the two expressions: S n - 1/2 S n = 1 - 1 / 2 n+1 or equivalently, solving for S n: S n = 2 ( 1 - 1 / 2 n+1) But now S n is a simple sequence, for which we know how to take limits. In fact, from the last expression it is clear that lim S n = 2 as n approaches infinity " Out of a sudden, I remembered that I was told in primary school "The addition and subtraction is allowed ONLY for series proven to be convergent (series which are converging to a finite number as n goes to infinity)". I would very much like to know what do you think about this? Regards, Alireza Bab-Hadiashar, Department of Electrical and Computer Systems Engineering, Monash University, Clayton, VIC 3168, Australia.

- 379. Michael Chisholm
- This site looks like it could be a real help to us struggling math students! Real analysis is a difficult subject to learn. I was looking for help with Riemann integrability though, and that part isn't finished. Ahhh welll....

- 378. Tim
- I believe the first term should be 1/2 and not 1, since the series starts when n is equal to one, not zero. Thus the Sum of the series should be one, and not two. .

**(Correct - has been fixed, thanks a lot; bgw)**Thank you for your time. Yours, Tim Maull - 377. Ken
- Prof, Your Interactive Web Site is a great idea, well organized and one of my favorite places on the internet. Your text is a great supplement to the course I'm taking now. It was highly useful last semester BUT... ...When we will see the sections on Integrals and series? Thanks-Ken

- 376. Linda Silverman
- Great page. I'm sending the address to all my students.

- 375. marshall
- Dr. Wachsmuth - I am currently putting together a syllabus of IB Higher Level Mathematics for the Web, my effort so far being at http://www.hk.super.net/~cismath/IBH.htm. I have already mentioned "Interactive Real Analysis" (see the page on Functions and Calculus) and am now asking your permission to use some of your gif's. I am thinking particularly of incseq.gif to illustrate a monotone, increasing sequence, which could illustrate item 4.2 of the optional material on the syllabus. Any use would be fully credited and stricty for non-commercial use. Yours truly, Paul Marshall, Head of Mathematics, Chinese International School, Hong Kong

- 374. Moran Gavish
- 373. Byron Hale
- What a great idea!

- 372. Mauricio de Castro
- Dear Sir, I am professor of Math in a second rate university on Brasil(Centro de Ensino Unificado de Brasília - CEUB). I saw your "Interactive Real Analisys" and it seems wonderfull to me. I would like to get one copy, perhaps in CD-ROM. Is it poss ible? I would like to say many things about your program, but my english is very poor. Would you read portuguese?

**(I wish I could - but I can not; bgw)**Many thanks Oltáchio Mariano Carneiro - 371. Berj Bsnnayan
- An excellent resource! I wish more courses had references like this!

- 370. Ira Mantz
- 369. Geoff Hagopian
- Pretty durn neat! Spelled "zoon" in family plotter.

- 368. Alfonso Ramirez
- First time I enter this page, and find the book like an excellent potential for supporting analisys courses, like for myself. I am teaching graduate students in educational mathematics in National University of Mexico. Congratukations.

- 367. Oltßchio Mariano Carneiro
- 366. Joe Krueger
- The last time I looked and thought about real analysis was in 1963. And the little I've had the chance to look at on your site looks interesting. There's one slight problem, however: I'm unable to get to the material on the integral. Maybe it's because I have a WebTV. The problems I can look at are interesting. Thank you, Joe Krueger

- 365. martin
- GREAT work!!! I like it so much. It is what i expect on net. BTW may I download the book to my personal computer? THANKS for the nice job.

- 364. Elton or Bonnie Lacey
- Bert. I am professor of mathematics at Texas A&M, I wish I had discovered your on-line real course last term when I was teaching real analysis. I am teaching abstract algebra this term. Do you happen to know an on-line source for this? Elton Lacey, 4058 Deerfield, College Station, TX 77845, 409-774-7065

- 363. mark tillinghast
- Beautiful. A vision of delight for anyone interested in Mathematics.

- 362. Lloyd Martin
- I am pleased to see that someone thinks advanced math can be taught by distance learning. Keep up the good work!!

- 361. Ryan Morris
- Hey guys, really great! I found this really handy. I'm 15 and studying advanced calculus. I'm really into the mathematical aspects of physics,especially particle physics. One small plea to you guys, could you at some point in time add a section on Fourier Transforms, that would be so great. I'll definitely be coming back here.

- 360. Lee Siew Tong
- Dear Sir, I am currently a third yr student majoring in Mathematics in National University of Singapore in Singapore, a country in South East Asia located near the equator. I've read through some sections of the interactive real analysis and find them interesting and enriching.Currently, I am taking a course in applied analysis whereby the content covers Normed spaces, complete spaces and lebesgue measure and integration theory. I will introduce this website to my friends who are also interested in mathematics. After reading through the sections, I have a question in mind: Can the materials that appeared in this interactive package be downloaded? Is there any copyright or any subscription before one can make use of the materials? If there is , can I be enlightened as to how to go about getting permission to download the materials? I look forward to your favourable reply. Thank You.

- 359. Malcolm S. Turnquest
- 358. Arndt F. Laemmerzahl
- Your site is proving very useful. Many thanks.

- 357. mrFrank
- your site is beautiful you're maths is out of my league I want some algorhythms for interesting movement for graphical objects. Sine waves are about as far as I have got any suggestions? you're great grovel etc.

- 356. Wilman Brito
- 355. Andrew Wrigley
- 354. John W. Dooley
- I sent my freshman physics majors on a web search for Zeno's paradox, and thought I had better check what they might find. Your development for the geometric series is transparent and pleasing... better than I would have done. Thanks for putting it up, John W. Dooley, Physics Department, Millersville University.

- 353. Jimmie Lee Johnson
- Looks very good. Recently (28,30 Jan) I have had some trouble getting through to you. I have recommended it to my students in my senior Real Analysis course.

- 352. Jonathan Picklesimer
- Incredible book! I am extremely pleased to find such a wonderful resource on line! Keep up the great work!

- 351. Larry Smith
- What you have done is outstanding! This type of use of the internet for presentation of mathematics is precisely what I have been hoping to do. I see I have a long ways to go, but I can see that it can be done. Thanks.

- 350. Philippe ThTvenaz
- Murphy's law. I came for an explanation of Lipschitz-alpha continuity... and it is exactly the point where your tutorial actually says: "Under completion, be patient!" But I nevertheless enjoyed all the rest. Detailed comment: Function primer, lemma within countably many discontinuities, should not the intervals [0,N] be replaced by [0,K]? Also, shouldn't "in total, at most ... or equal to K." be replaced by "... to M."?

- 349. albert van loon
- 348. Molli Bowen
- 347. Dave Cohen
- 346. Robert Sharpley
- Just getting started looking at your page, but will be checking back on it from time to time.

- 345. rmurray
- 344. Cheryl R. Hepp
- I am a senior at Roosevelt University in Chicago. My professor (Jimmie Johnson) introduced our class to this web site with the recommendation that it would be a greeat help in our study of Real Analysis.

- 343. Lyle Barbato
- Wow!, all I can really say is wow. This place is truly the best math resource I've found on the Web. I want to thank you for making it available. Thanx!

- 342. Amanda Prest
- A very good idea. I will definitely be using this page again.

- 341. Jimmie Lee Johnson
- 340. christopher weldon
- 339. Alan Cooper
- 338. Michael Banther
- This may be a Godsend. I'm starting a course on H-infinity control and finding out that I do not have enough Linear Algebra under my belt to really understand it. I found Interactive Real Analysis by searching for Cauchy Sequence in Yahoo. I will definitly be spending some time here in the next several days (and weeks probably).

- 337. Carlos Arturo Alatorre Machuca
- I really love this kind of themes and I'm so proud seeing that people as you is interesting too. Most guys prefer to watch some "Sex's site" before consult this pages and I think they are wasting their time. I'm 23 years old and I congratulate you of your work. P.S. Excuse my english, but understand that without my dictionary I could'n get it. Bye, and thank you

- 336. bobot fabrero
- I wish to see more of Lebesgue integrals in the undergrad level. Also more unusal subsets of the real line. You have done a superb job in making this homepage.

- 335. Lisa Metzler
- This site was recommended by one of my professors. Great resource!

- 334. Jamie Wallace
- Very interesting

- 333. John Tankersley
- Hello, I was looking at the web pages Interactive Real Analysis. Do you know when you will be finished? I think you are doing a good job on the project. Thank-you, John Tankersley

- 332. Henry Power
- Just started using - really enthused about using.

- 331. Syed N.U.A.Kirmani
- Great Job! Many thanks.

- 330. Serna
- Thanks for providing such a wonderful service.

- 329. Carlo Santarelli
- 328. Amit Kagan
- Marvelous! The best math instruction I've seen on the web, I hope others sites will follow...

- 327. dimitris gavalas
- 326. Antony F Ware
- I have browsed with great admiration and enjoyment your interactive real analysis course. I am about to teach such a course here in Durham (UK), and will be pointing my students in your direction. The only difficulty is one of speed---sometimes the transatlantic traffic can be pretty heavy. Would you consider the possibility of setting up a mirror of your site here in the UK? I'd be more than willing to host it if so. Yours, Dr. Tony Ware, University of Durham, Department of Mathematical Sciences, Science Laboratories, South Road, Durham, DH1 3LE., U.K.

- 325. Marco Zavarella
- Your work is simply wonderful. It's the first time that I see a nice work like this. Please continue in this way. I thank you very much !!!!

- 324. J. Edward Orbin
- 323. Per-Erik Bemer
- I think this is a vey good page on the "net" because it has a sorce where they have got the information from. Best regards from Sweden, Stockholm,KungsSngen

- 322. Joaqufn D'Opazo
- I am waiting for Metric spaces and the Integral Thanks to all of you for a very good work

- 321. Fava,Alberto L.
- 320. Patrizia Carnesecchi
- Molto interessante!

- 319. Partha Dass
- I was looking for some info on Lebesgue integrals. Definition, how is it different from Reimann sum etc. Looks like you are still working on the page. Let me know if you have any info on that or atleast if you could give me some references. Thanks, Partha Dass

- 318. G.Andersen
- 317. J. Waldron
- I think this will be a tremendous resource for students!

- 316. Mickey Di Siena
- 315. Lisa Greenberg
- Great application!

- 314. Keith Alexander
- Excellent, comprehensive resource discovered serendipitously, while surfing through mathematics-related sites on the Internet. Most recent link: Cantor. Grad. student of Computer Science at Wayne State Univ., Detroit, MI

- 313. Don Coupe
- Great resource, one problem. My browser (Microsoft Internet Explorer 3.01) gets stuck on the title page and I have to shut down. Otherwise seems OK

- 312. Chris de Vreeze
- A superbly constructed site. There should be more of these! My compliments.

- 311. maganti, lata
- fantastic

- 310. Chris Cannon
- Came looking for an explanation of Zeno's Paradox - liked it so much I'm going to have a good look around

- 309. Christopher Spiteri
- Anything better than "brilliant" would do. I'm reading B.Sc.(Hons) in computer science and this site simply covers all the material there is. Keep up the good work

- 308. David P. Campagna
- Interesting project -- I can't wait to check it out!

- 307. Mark Falco
- This web page is well done and informative. Recently I needed to review some theory on partial sums to implement a DSP logic circuit, and was pleasently supprised to find what I needed. By the way,The Map function is great!

- 306. Andrew Bunn
- Sweeeeet!!

- 305. Ramin Tavassoli
- I have skimmed through some of the topics on real analysis. They are treated quite well, and the conceptual treatement of some areas supports the strictly axiomatic methods commonly used nowadays to introduce new subjects, which has the disadvantage of becoming too abstract. The treatment here leads to a good and quite detailed understanding. I hope this will continue, I will come back and use this site more often in the future.

- 304. Benjamin Shaw
- Nice work. Do you know any algebra/group theory sites like this?

- 303. Burt Shaw, P.Eng.
- This is one of the most rewarding web sites I have seen recently. As an amateur mathematician I commend you on doing such a fine job. I look forward with anticipation to your completed chapters. Sincerely, Burt Shaw, Director of Systems, Juniper Lumber Co., Juniper, NB Canada

- 302. Warren Oliver
- 301. Marcus Hobbs
- Hi Burt, I just wanted to let you know that i really enjoyed your web page. Marcus, Walt Disney Feature Animation

- 300. Jim Ratschler
- I really like the structure of the site and find the whole idea very compelling.

- 299. Edwin Hensley
- Thanks for the info!

- 298. Eric Reither
- GREAT pages, I'm in 11th grade and have been challenged with proving Euclids Theory e^(i*pi)=-1. Didn't find the answer but now know of a great source for other information. Thanks

- 297. Don Coupe
- Your not going to believe this.. This is exactly what I am looking for. This morning I misplaces my copy of Kolmolgorov (Introductory Real Analysis)

- 296. David Tomb
- 295. Delfim Cardoso
- I'm pleased to see your service being provided to everyone at no charge on the world wide web. While many of us in university struggle with new concepts and theorems in mathematics, it's helpful to have a place where you can look up a certain topic. The way things are laid out make it easy to read, and often explained better than what I've seen in most textbooks. Thanks! :)

- 294. K. Baker
- 293. kyle kirkland
- 292. Margie Hale
- 291. Stanley Cohen
- It's an interesting work. I have a little trouble reading the typography. For example the f primes are hard to read. Also the examples in "Functions and Limits" are unclear. The vertical line positioning seems to be off. How do you read (-1) raiesd to the italic n/n power in example 3? By the way a clickable map would be useful. Incidentally, I came across you interactive text while searching the web for "Cauchy".

- 290. James Heddle
- Dear Bert, I stumbled across your Real Analysis pages while searching the net for information on Barker sequences and would like to congratulate you on creating an excellent resource. I am now motivated to brush up on my neglected pure mathematics. Thanks! James Heddle, Acoustical Consultants (Voice: +61 7 3822 1020, Fax: +61 7 3822 3200), Brisbane Australia, Web Site: http://www.gil.com.au/comm/jhacoustics/

- 289. Jan Morovic
- It's great to see such excellent use of the web - this site is definitely among the top 3%.

- 288. Dave Santucci
- One of my colleagues in the calc group of Texas Instruments, Dave Stone, saw your talk at the ICTCM.I am learning about Java so I was interested in seeing how your applets worked. I have Netscape 3.0 but have not gotten any to run so far. Will keep trying.

- 287. Simon Lockwood
- I needed a quick refresher on math notation for sets, functions and series - this hit the spot beautifully!

- 286. David J. Mieczkowski
- very well organized. Contains a lot of interesting information. You really need something on the Euler's constant gamma (not e, but the other one).

- 285. Patrick Malte Joseph Ingram
- Hi. I'm a student at Simon Fraser University in Vancouver, Canada. I just visited the online Real Analysis page and I like'd it quite a bit, and I have a question : how did you display mathematical symbol in hypertext? I'm working on a web page for our math student society, and I could see that this would be useful, but I have about as much skill and background with a computer as your average english major does with Fourier series. Thank you, Patrick Ingram

- 284. Harvey J. Greenberg
- Neat page. I can't wait for you to finish Interactive Real Analysis. Harvey Greenberg

- 283. Andrew R Sharpe
- I have been browsing about your Real Analysis web site, and while leaving a comment, noticed other comments by high school teachers. I am taking Real Analysis as a third year university course, could high school students really be taking the same thing? Are we at Mount Allison University really that far behind? I guess my question is, when is the usual level at which this stuff begins to get introduced? let me know, andrew

- 282. Tony Phillips
- Hello, Bert! I've been enjoying your interactive real analysis page. I'm teaching "Intro to Real Analysis" here at Stony Brook this term; I've put a linkto your text on the course web-page (http://www.math.sunysb.edu/~tony/ana/) and I hope the students are using it. Small error in introductory material on continuity: I don't think you should hint there that continuous functions will take DIVergent sequences to DIVergent ones. E.g. constants (duh). I didn't have much luck with the "continuity checker" ... the control panel kept blanking out. But it looks neat and I hope it can run on my machine soon (sgi indigo). I'll try it later on the PC's in the department. Best wishes, Tony Phillips

- 281. james a. black
- from college days now a practicing neurologist with a yen for analysis as an intellectual hobby... Very interested in when the integration section will be up and running. Also would like your recommendation on good intro to measure and integration for someone versed in fist 8 chapters of Rudin's Principles. Thank you, Jim Black

- 280. David Styer
- 279. Benoet Favreault
- I find your text-book very convenient for me, a graduate student (in semiotic at Universite du Quebe a Montreal) whose competances in math are close to zero. I was just looking for some explanations on the concept of continuum as it appears in Ch.S. Peirce semiotic. It is presented as an application of Cantor's conceptions. Well, It is pleasant to find explanations that I can follow without feeling like a perfect idiot (wich was the case in somme foudational texts) I am not sure my understanding is correct, but I have the feelling of acquiring something new and useful for my further researchs. Excuse my English, This is not my mothertongue wich is French.

- 278. Zachary Uram
- Wow fantastic page! Do you know of any topology web pages or pages on non-standard analysis (ie Robinson et al..)

- 277. Bruno Stucchi
- 276. Mark Hoskins
- With this Web page, who needs Royden's text? :-) I'm an OLD man (50) who's returned to school for an M.S. in math, and find your page very interesting, enjoyable, and useful.

- 275. Karen Caldwell
- 274. Kartik Subramanian
- Could you reccomend some good, entry level books for a Course on Real Analysis ? I found the page very interesting and the concept is also very good. For a graduate Electrical Engg. student like me, I suppose these topics are very essential and useful, Thanks for the same,...

- 273. Dina Roumiantseva
- Thanks!!! This a really great idea and it's quite well done! Very involving and informative.

- 272. Antonio Casanova
- I whish I had this tool when I was student!

- 271. Autumn Bahr
- I think that this is a wonderful help for CalculusII at the University of Idaho... Autumn Bahr

- 270. Karen Saxe
- 269. sabine g vogel
- this is an excellent online tool for math students. please finish text and create others !

- 268. Patrick Ingram
- Good work! I like this project quite a bit, and it looks like it could end up being a very useful reference.

- 267. Phillip Goss
- This page has proved very valuable to me. I am doing a course involving set theory and the material here is just right. Precise, well formatted, I can see the hard work behind it ! Keep it up folks !

- 266. David Kelley
- This is one of the best sites I've seen on higher calculus. Keep it up!!!

- 265. Kevin Beimers
- I just finished an essay on Zeno's Paradoxes and i footnoted you guys. Thanks for your help! -kev.

- 264. Cara Fabian
- I have an intro to Anatysis class now. This page has been very interesting and helpful.

- 263. natalie moulton-levy
- you guys are really cool and have helped me a lot! keep up the good work. I like to see these kinds of productive web sites which can actually help people in real ways.

- 262. Kelly Crain
- 261. Andrey
- I am currently studying Calculus and find this interactive document to be of great value. Is it availabe in other formats? DOC, PDF? Thank you. Andrey.

- 260. Michael Butler
- Nice site and relevant for my studies in discrete mathematics

- 259. John Egenolf
- A colleague just told me about your site. This is my first look. I think it is a good idea. Will have further remarks after I have a chance to review it a bit.

- 258. Gustavo de la Rosa Muruato
- 257. Brad Tressell
- I am going through an analysis class right now and your page is very useful.

- 256. Massimo castagno
- I am a researcher in a telecommunication firm. I love analysis and I think all you have made a great job! Thank you!

- 255. Misha Kapushchevsky & Peter Haberlandt
- Cool. Man, this online text book is way cool, beyond any imagination. :) I'm taking Math 413 at Cornell, in essence the same course your textbook is designed for, and I ran across it in a random search on the web -- I'm absolutely impressed, astounded and excited.It is absolutely great how you did the typesetting on this - small images are well-done, it's grand - btw, did you know that IBM released a TeX plug-in for netscape, so in theory, perhaps in the future, stuff like this will be possible to type-set in TeX and put on the web immediately. As a comment, your Java tools are absolutely amazing! Pete and I we are still wowing. Oh, it does not work in Internet Explorer 3.01, for some reason. Did you have any personal quarrels with Billy Gates? :) Any plans for making an online textbook for complex analysis? Well.. Thanks for the great resource! P.S. Here's a suggestion, we found this missing, in your presentation of Algebra on series, it'd be really COOL to have a counterexample for the term-by-term multiplication of series: if sum(x_k) and sum(y_k) are convergent, then sum(x_k y_k) is not necessarily convergent, say for x_k=y_k=((-1)^k)/sqrt(k)... This was also missing in the textbook we're using and gave us headaches. :) Anyway, again, this is just way too cool. "It was cool." --Pete.

- 254. Bedir Tekinerdogan
- A very useful home page. Others should follow such a course approach on the net for different topics.

- 253. Jason Petry
- Excellent Resource. Great Help in showing some convergence texts for series not covered in my Real Analysis Class textbook. (Abels, Root Test)

- 252. Deborah Van Vlack
- Thank you for creating this site. Yours are some of the best explanations of mathematical concepts that I've seen.

- 251. F. Bradley
- Best site I've encountered in a long time! Keep up the good work!

- 250. Janie Levine Hellyer
- We'll pass this one along to our readers! Janie Levine Hellyer, Editor Family Learning Exchange, PO Box 5629, Lacey, WA 98509-5629

- 249. Hiren Gajipara
- 248. Lewis Park
- WOW !!! this sight make learnig fun and practical. May hat goes off to the guys/gals who created this web site. Being a Computer Science /Pure Math Joint Major one can truely appreciate good work when one see's it.

- 247. David De Benedictis
- 246. George Salazar
- 245. jamal Oulhadj
- This interactive book is one of the best ideas i have seen so far. I hope, resources permitting, you will extend the topics covered i.e Linear Programming, Partial Diffs,...

- 244. Thomas Gunnarsson
- 243. MArgaret Heinrich
- I am currently taking real analysis and am thrilled that someone finally thought to do this. It's difficult to find anyone to help in this subject.

- 242. Judith
- 241. Danielle Driscoll
- This site has been a tremendous help for my Advanced Calculus course. I am anxiously waiting for the unfinished chapters. Do you have any suggestions for help in the meantime?

- 240. Omar Khan
- This is a neat idea. By the way do you have any links with the real IRA.

**(I have absolutely no connection to the 'real' IRA; bgw)** - 239. Ed Robison
- 238. Najmeddine Mohamedm
- Just browsing around very briefly, I would say that I found it very interesting. Given the fact that I 'm a physics major, I 'm wondering whether there is such thing for physics books.Thanks. Mohamed.

- 237. Dave Erwin
- A great step forward, and - in general - well implemented. I really like the instant feedback from the exercises. One problem I have noticed - `closure' is not in your glossary

- 236. Mike aka Plant 42
- I have been able to find what e is and how it is derived, but I want to find more. I am trying to find out what he was useing it for. What was reason that he was looking for it? Is there a pratical answer like he needed it to build something,or was it just found for fun found in his spare time? If you have any idea to my answers, I would be greatly pleased:) I have looked everywhere I can think of. Thank you for your time. Mike aka Plant 42

- 235. Lisa Beck
- I find the contents very helpful for my Advanced Calculus class that I am taking. I am anxious to see the rest of the chapters once they are completed. Keep up the great work and thanks.

- 234. Lachlan McDougall
- Thanks for the info. I have used it for my project. Keep up the good work.

- 233. Kristen R. Wottreng
- 232. Andrew RW Sharpe
- Thank you very much for a making such a helpfull site. Yours is the best Real Analysis web site I have come across yet. I imagine I will be here very often during the next semester. For now I'm off to read about Perfect sets. Thanks again, Andrew Sharpes

- 231. Kayhan N. Etemadi
- 230. Adam
- This is a great tool for Real Analysis

- 229. Dr. Zong Zhi
- It is interesting and easily undertandable. I think it would be more interesting if more pictures were included.

- 228. Pedro Tytgat
- Waow! This project (or rather : it's realisation) is really impressive! I'm a math's teacher. This really opens new perspectives. Congratulations to the team that set up this site! Pedro Tytgat, Belgium, Europe

- 227. Danny
- I am glad that I found your side on the web. It gives me some useful explaination on what I am currently learning in my class-elementary analysis. If you can put more problems(difficult ones) and solutions on it, it would be much better. Looking forward to see your side improve.

- 226. Harry Ashworth
- Bert - what do you use to create your .gif files for the equations. Since Word is disappearing from Mathsci, I need to put a whole bunch of Equations ffrom Word 2.0 (Equation Editor) to html. The WP6.1 converter will not read Word Equations so I can't have the students use WP to look at my Word files anymore. I've used Corel Transit to convert Word stuff to HTML and it does OK except it can't handle PI, symbols etc., obviously, so I guesss GIF is the only way. Do you have a neat trick to go from existing equations to .gif?

- 225. Douglas McCarty
- 224. Lou Blazquez
- My seventh grade honors class is doing research on mathematicians (hopefully, that are still alive so they may contact them). Do you have such a list? Do you have their ethnic backgrounds listed too? Lou Blazquez

- 223. Emrah AKYAR
- I like it... I m a research asistant at mathematcis department I gave your address to my students...

- 222. Jeff E. Lewis
- 221. Jeff Kemper
- I teach math at Helix High School in the San Diego, CA, area. I am taking an Analysis course currently at National University, and what I see here I am quite impressed with what is available. I will tell the other students about it. I hope that this Interactive Real Analysis page will continue and that an example problem area could be created. Thank you very much, one and all, that worked on this web page!!!

- 220. patrick stevens
- I'm curious as to when the rest of the chapters will be finished.

- 219. Ramon Vilageliu
- your web is a fantastic tool

- 218. Paul Clark
- Great site - thanks!

- 217. Brian A. Viscusi
- good idea!

- 216. Emily Proctor
- What an amazing site!

- 215. ernesto somma
- terrific stuff!!!

- 214. Arumi Martinez
- I think this is great!! I,m doing a Master in Educational Maths. When I found you, I was quite happy for a chance to study in a fun and useful way. CONGRATULATIONS FROM MEXICO. Arumi.

- 213. Chris French
- This is an excellent site! I've been waiting for the likes of it for some time. Thank-you! No more overdue books at the library whenever I have a question. I do have one request: could you include sections on the theory and use of Laplace transforms? Theory and application of Fourier series and transforms would also be helpful. Thanks again!

- 212. Vladimir Livshits
- Hi, you have a really cool page here. We're trying to do something similar, so you might want to check out http://liconsortium.cs.sunysb.edu and my personal page at http://mathlab.sunysb.edu/~livshits/licil/LICIL.html. It has some Java stuff on it.

- 211. Brenda Norman
- I love your tutorial. It has been most helpful in my studies. Thanks!

- 210. PESCH Gilbert
- 209. Michael Kilkenny
- 208. Donna Sutliff
- I appreciate this website very much, especially since it's been a long time since I've studied math. It is very well organized!

- 207. maurice de gosson
- 206. Larry Iles
- 205. Lee Blankenship-Brown
- Thanks for making such a well-organized study tool available. Is there an analagous web page for abstract algebra?

- 204. Cynkris
- Thank You!!!! I have been searching for information on Johann Bernoulli for a report my daughter has to do and had almost given up hope in finding more than two lines about him. Then I stumbled on you! Because of your indepth article she now has more than enough information to write an A report on him.

- 203. Scott Person
- 202. maurice de gosson
- great !

- 201. Kevin Naik
- Way Cool informative site !!!! Definite plus for Maths Students

- 200. Jim Taylor
- Splendid! Thanks...

- 199. Eric Andersen
- I am currently taking real analysis here at Brown University with Professor Andrew Browder. These pages seem like a great reference tool. Thank you! Eric Andersen, Brown University '98

- 198. robert bjorkland
- 197. Marco Lizza
- These pages are really interesting, since they are a compact and quick (but exaustive) reference... there's only one thing to notice, that is the lack of the theory about (I guess the names as I don't know them in English) MULTIPLE INTEGRALS, PARTIAL DERIVATES, DIFFERENTIAL EQUATIONS, and so on... I suggest to include also these topics in the current program.

- 196. Alejandro Samayoa & Francisco Canet
- Thanks for helping us solve a homework that we had for our Calculus I course on Industrial Engineering here in Guatemala C.A. on Rafael Landivar University.

- 195. Michael Weingart
- 194. Marvin C. Goff
- Looks great so far. I'll comment later after I have studied the project more.

- 193. Jerry Rieper
- WOW!

- 192. Roger DeBar
- Excellent work. Congratulations.

- 191. Fabio Brugnara
- It is an excellent idea to provide reference material on-line for commonly needed topics as these. It is also very well realized. I hope you will provide similar resources for other fields of mathematics as well. Thank you.

- 190. Glen Davis
- Interesting site. Are there links to others of the same type?

- 189. Erik M. Simpson
- I am looking for information on metric spaces but I can't seem to get your metric page to go past the index. Please look into this. Thank you! This is an excellent resource!!

- 188. Alexander Bogomolny
- It was a real pleasure to leaf through your book. The interface is well thought off and consistent. The material is in abundance and well presented. The access and search is easy. The Java applets are admirable. You are welcome to visit my page: http://mars.superlink.net/abogom/. Keep the good work, sincerely, Alexander Bogomolny, Ph.D.

- 187. Honeycutt,T L
- 186. Shawn Legrand
- 185. george best
- 184. Dale Collins
- I have just found the site. It looks interesting. Do you have any ideas of a Web site or a usenet group where I can post a question about the Weierstrauss approximation theorem?

- 183. Tarra Soeng
- Nice pages, interesting grpahics. I was interested only in Zeno's Paradoxes, some info of which I will be using in a philosophy paper. Thank you!

- 182. Brina
- Hi, I am a Taiwanese student in England. Seldom to see such a nice home page as yours in Taiwan. If you like, you may also visit my home page: http://www.csv.warwick.ac.uk/~aepbj

- 181. Jerome E. Hunt
- 180. Anand Kuratti
- This site seems like a good way to present fundamental mathematical concepts. Although I am not a mathematician, I find it necessary to acquire a good mathematical background to complete my research.

- 179. Donald F. Ryniker Jr
- I LOOK FORWARD TO ALL RYNIKER/RINIKER EMAIL.

- 178. Mischelle Merritt
- 177. Larry T. Lenston
- 176. Abdulmalik Arif
- Dear sir, For a long time i was looking for an on line distance learning subjects specially mathmatics ,and it seems that i found my starting with your's Real Analysis.

- 175. steve kim
- 174. oscar jesus perez
- opinion(in spanish): excelente libro. Lastima que no pueda encontrarse en mi pais.Es de gran interes para un curso de analísis en la la facultad de ciencias universidad nacional autonoma de méxico.

- 173. stuart leviton
- I look forward to studying this course. I would like to learn as much mathematics as possible. Respectfully, Stu

- 172. Tim Cranny
- Very nice work!

- 171. Kenton Brown
- This is a wonderful resource that I just discovered. I am currently a junior at Yale University where I studied real analysis with Prof. Serge Lang last year, by whom I was well schooled in the rigorous theory of this subject. Little did I realize that when I wanted to remember his proof of the abel theorem of convergence (together with the summation by parts lemma), all I had to do was check your site on the world wide web. Keep up the good work, you can be sure I will return often.

- 170. Gregory Guglielmetti
- 169. Cobi Callev
- i'm currently studing for my BSc in math in Tel Aviv University (israel), and i got to say that u'r interactive book here is gr8! it could use some inproovment, like filling in the missings prooves, and maybe giving more that 1 proof for some important theorethms, anyway -- gr8 work! i'd love to see pages about complex analysis and advanced algebra too

- 168. Kazuaki Nakanishi
- I hope that you'll print portraits of mathmaticians.

- 167. Kirk Nussbaum
- 166. Jorge Torrez
- its a good start for self-students in this interesting field of the mathematics.

- 165. Viktor Tchistiakov
- Nice work, very handy environment. Viktor from Applied Mathematics (TW), Home page (TW) |

- 164. James Thomas
- 163. sugato chakravarty
- I am not sure that I see the "interactive" part in "Interactive Real Analysis." Interactive to me suggests reader participation in some way. I do not see that here. There is a bunch of stuff that one can scroll through and print out. Is that the extent of "interactive"?

- 162. Joachim Huber
- exelent intruduction in analysis!beeing an architect working on a dissertation about "urban topology" i'm looking for topologists interrested in logic and philosophy of science. Even i'm an autodidact i'capable of reading stuff like bourbacky 1+2. Right know i'm searching for material about current results in "surgery", maybee written in an easy not too formal language. For any hint I would be thankfull cause working as a research assistant in Weimar in the ex-GDR, it is extreamly difficult to communicate on an up to date level. thank you! your faithfully joachim huber dipl. architect.

- 161. Jeronimo Pellegrini
- Very nice idea! I hope there will be other books available soon. J. Pellegrini

- 160. Brian Junker
- 159. Howard Seltman
- Just took a quick look so far, but it looks real interesting.

- 158. Hans Kestler
- Very good so far. Helps to look up topics once learned.

- 157. harry ashworth
- Very cool.

- 156. George Fisher
- A very interesting approach to delivering information on the web, as well as a real help to me with a problem in set theory at the moment. Thanks for putting this together!

- 155. Gustavo de la Rosa Muruato
- Great idea.

- 154. Rachel Garden
- 153. Jonathan Paige
- 152. MERK
- This is great for undergrads I suppose. But what would be really cool is something like this at higher level, lets say at the level of Gerald B. Folland's "Real Analysis". Your course ware could improve on his book by providing numerous detailed examples of all the concepts.

- 151. David Robinson
- This is a wonderful site. I enjoyed reading your histories as well as your textbook. Thanks a lot.

- 150. Jean-Guy Chamaillard
- Hi! I was looking for something in Calculus and I found the right place. My students will surely be interested by some of the material I've read in your sight. Thank you very much, Jean-Guy Chamaillard, Kirkland Lake (Ontario)

- 149. Pete Peterson
- 148. Chris Barnes
- 147. Alan Peters
- This is an excellent academic service. I have found having the the fundamentals of real analysis accessible over the web to be extemely useful. Thank-you for making these pages available.

- 146. Robert Mitchell
- 145. Ray Grinnell
- Quite an interesting and informative project. I teach advanced calculus here in Barbados and the course has a fairly large component of real analysis as covered by your site. It may be quite useful for my students to use your site for the next semester in the fall.

- 144. chan kok hoong
- This is the most wonderful books that i have ever read.

- 143. cherae dejean
- 142. T. Zachariah
- I just visited "Interactive Real Analysis". I am interested in looking at these pages in more detail during the summer. -- Thomas Zachariah, Loyola Marymount University, Los Angeles

- 141. Yuan che
- 140. Intercom Online
- Intercom Online

- 139. Irina
- Hello! I am Irina. I live in Russia.I am glad to meet with you.

- 138. Kim Hamilton
- This is great. I havent seen much mathematics at this level on the net. Id really appreciate knowing where to find more of this, or any other sites which contain higher level mathematics.

- 137. Otto Pulkkinen
- A very useful reference

- 136. Julie
- 135. sugato Chakravarty
- Looks like a great venture...I shall certainly be following its progress with interest

- 134. Steve Arnold
- Your site was great for the calculus project I was working on

- 133. john carter
- 132. Frank Burroughs
- Late night cruise hunting for more info on a course I'm entertaining.

- 131. William C Davidon
- Thanks for making all the 1k GIF's available. Please let me know if (NOT EQUAL) is somewhere at your site. I'd like to use it but couldn't find it.

- 130. Bernd Krause
- 129. J.P.Lewis
- 128. Jason Simpson
- was really impressed with the information on this web site.

- 127. Michael Fitzpatrick
- This is a great site. An excellent source of information !!

- 126. Ewout Verkerk
- This is a very handy reference for mathematics. A pity that at the moment the integral-page is not accessible.

- 125. Bill Moore
- Great idea- I like the historical sketches. Please continue the good work!! I noticed that some icons did not "click" - are these "still under construction"?

- 124. T.M.Bowles
- 123. Mats Magnusson
- 122. Ubaldo Pernigo
- Good.I was looking for something like that

- 121. Ted Albright
- Good start, I question your proposition on rationals are countable. Though it is true using your definition but is it a universally accepted definition. It seems the truth of your proof is based on your definition. Good work.

- 120. Nils Bruin
- Nice to see this kind of experiments

- 119. Tom Archibald
- Nice design.

- 118. jim black
- Good course...waiting for more on integration

- 117. Stephen Brick
- Very nice !!

- 116. Damacio Barbosa
- 115. Fernando Mercado
- Can you please post some of the proofs to some of the problems. This way, I can see if I'm on the rigth track. Thank you.

- 114. Rob James
- Nice use of the Web! It shows what an excellent resource it could be.

- 113. Mikko Haapanen
- This IRA ;-) is just great!

- 112. Tim Rossiter
- Nice Job

- 111. Harry D Jeffery
- Significant. I love something I can wade in. Thanks.

- 110. Mario Lemelin
- Very interresting concept. I wish that I will be able to do the same for physics. It depand on a grant I hope to receive. I think that this is very generous and great to offer such a place to help student understand.

- 109. Yoshikazu Kobayashi
- 108. Leon Dague
- I don't have any comments yet, I just got here.

- 107. Gilles G. Jobin
- Please excuse my very poor English. I was visiting your site and I just want to drop a word to congratulate you. It is a wonderful mathematical site. Your use of frames is very good! All the best, Gilles Jobin

- 106. ONG TEN SWEE
- I am impressed by what I have seen. Very informative. Unfortunately the chapters on the Integral and Metric Spaces aren't available yet.

- 105. PATRICK BARNES
- LOOK GOOD ! ENJOYED MY VISIT

- 104. Virgil C. Smith
- thank you so much for this review/tutorial. It has helped me a lot to get an handle on the introductory material in real analysis.

- 103. stephen watson
- This is a beautiful site! Congratulations. Please have a look at http://www.unipissing.ca/topology/

- 102. Brandon Gage
- sfkl;sdf

- 101. Steve Lay
- I haven't looked at it yet but intend to do so soon as I am interested in tool development for projects of this nature. For more information about the research group of which I am a member see the following URL: http://www.amtp.cam.ac.uk/icrd/

- 100. Joocheol Kim
- Excellent approaches for studying real analysis

- 99. dorothy
- 98. Eckhard W. Hellmuth
- Interesting method! Could you do a simple introduction for Chemists? We need something for beginners.

- 97. simon teague
- 96. matt padilla
- Wow, what a cool resource on the net. please keep it up! I don't suppos this exists in written form anywhere?

- 95. Feri Farassat
- 94. Diann Cashell
- 93. No Name
- I stumbled here and I am very impressed. Although the majority of this is beyond my math background (Calc I) I am researching set theory to solve lottery wheeling problems. Right now people "wheel" numbers by hand so there is no guarantee that a "cover" has been minimized.

- 92. Peter Kruus
- The info on geometric series was helpful. THanks.

- 91. Matthew Kilpatrick
- This online text is great. The only drawback is that the answers are not available for some problems.

- 90. steve harsh
- 89. K. David Prince
- Just got here. I'm looking for resources for students in our Minority Science & Engineering Program, University of Washington, Seattle WA. This site looks very good. And, I intend to let our students know about it.

- 88. Sunil Pinnamaneni
- 87. John Felix
- The best elementary analysis info I've found on the 'net. Please complete the "Metric Spaces" and "The Integral" chapters soon!

- 86. sam tannous
- Very nice. And complete. I'll be visiting your book often. Do you have any plans for other books? May I recommend one for Abstract Algebra?

- 85. falcon
- Nice, funny, and wonderful.

- 84. Josh Gordon
- I just discovered this today, and used it to verify that a certain dictionary definition of a Cauchy sequence is incorrect (or, at least, poorly stated.) Thanks so much for this service to the Web. What a fine service to the world this is!

- 83. Joseph Talafous
- Very excited to start. Trying to learn graph theory for chemical structures

- 82. A Min TJOA
- It is the best math page on Internet

- 81. Thomas
- 80. Hiren Maharaj
- EXCELLENT! I especially liked the `historical titbits`

- 79. Joseph E. Brandelik
- The material would lend itself to CD ROM for student studies.

- 78. Ekim Sahin
- 77. Jason Roth
- 76. Todd Holloway
- A real analysis online tutorial is a most welcome addition to the student looking for "that extra information"...namely me. Thanks... pier ps: there is a error on /projects/reals/cont/limits.html under the heading of One-sided limits of a function.

- 75. James Dildine
- 74. Massimo Castagno
- Great idea! I hope you will include all the issues in the present list and then increase database with other subjects (multiple integral, complex analisys and so on). Thanks for your initiative. Best regards, Massimo Castagno

- 73. Adam D. Jansen
- 72. Rachuri Sudarsan
- Is it possible to provide a Postscript version of this tutorial. The tutorial is very well planned and done extremely well. Thanks and Regards, Dr Rachuri Sudarsan, Computervision R&D India, SAI CHAMBERS, WAKDEWADI, PUNE 411003, INDIA

- 71. William Gordon Ritter
- I am a student in Honors Analysis at the University of Chicago, the hardest undergraduate math course in the nation. I think this interactive textbook is a great thing. It presents the basic concepts in a very clear fashion without sacrificing mathematical rigor or complexity of ideas. I think there should be more resources like this one available on the World Wide Web and they should be more widely publicized.

- 70. Birger Faxen
- I couldn't find any pictures or function graphs in the material. If there are any it is not obvious for the user how to see them. I think they are important in this topic.

- 69. Li-San Wang
- Nicely done! Can you include a page on bibliographies on real analysis ? With a short description for each book, please!

- 68. Neil Barnes
- Came looking for information on Metric Spaces, however that section isn't done yet... oh well.. Got this from WWW Lycos Search Engine (metric spaces), this was given as the second hit... Thanks for this sort of work, it's a very handy idea indeed.. Yours, Neil

- 67. Niloufar Darvish
- 66. Erik Saunders
- 65. David Cartes
- 64. Kevin Spradlin, Jr.
- 63. Mochamad Tito Julianto
- This is a great things. Wish I can create the same thing for another field in Math or Computer Science. Let me learn how to organize this excellent Online Tutorial.

- 62. Francesco Fontana
- I'll try to do my best in writing in english (is it correct ?) since i'm italian : I study Electrical Engineering in Pavia (Near Milan), and, yesterday I downloaded the capter that explains SEQUENCES. It is very simple and comprehensive and it's well organized.But today i tried to have a look at Sequences and .... of functions and i saw that there is no document in it. I suppose you are already writing them. By the way, i think your project is very ... cool (don't you say cool?) !! and it's the only one i find on the web ... go on like this, i think students will appreciate your work. Your faithfully, Francesco Fontana, Pavia - Italy

- 61. Anders Dahlner
- This is an interesting way of learning. Do you know any book of Functional Analysis in this way?

- 60. Jonathan Golan
- How much teaching success have you had with this? I have been writing something similar but don't see a major difference in student performances.

- 59. David Bindel
- I'm very impressed. I am a student in a course in Real Analysis at the University of Maryland, College Park. I am a double major in math and computer science, and I have a prior interest in computer-based learning resources, particularly in Web-based sites such as this. Not only does it look like it could be useful to me as a student, it is also very pleasing to see a hypertext teaching project like this one so well-implemented. Thank you.

- 58. Simo Kivel
- Interesting. The big question is: What do you get more in this way compared to a book? Hypertext links, of course, but how essential are they in this context?

- 57. jim black
- Just found this resource and am pleased to be able to peruse it THANKS

- 56. Dale K. Bird
- 55. Debbie Finucane
- Nice Layout, Did you know that the map button at Differentiable Functions in Limits Continuity and Differentiation, isn't working.

- 54. Katia Malinova
- I am a student of S.Petersburg State Uni, Department of Physics. However this year I study at University of Michigan. If you want to know more about Russia or my home University or just want to talk about nothing feel free to write me.

- 53. William Meisel
- 52. Anatoly Vorobey
- An extremely useful project. The level of presentation is sufficiently high, and the features of the Web are used nicely. The problems are a very helful addition, too. I'll probably be using the course. Thanks a lot!

- 51. liou yung-cheng
- 50. Bob Vallin
- 49. K.C. Reddy
- 48. Saifi Khan
- I found it wonderful and really very illuminating. Thank you very much. I am interested in recieving a copy of the roadmap of Interactive Real Analysis , in the form of a postscript(.ps) file. If possible kindly send to me. With regards, SAIFI KHAN

- 47. Kimyong Kim
- Exciting. I am working with lynx oriented text using texi and texi2html translation. I would like to grab some of your ideas to write discrete math.

- 46. Edward Reinke
- 45. Dr. David E. Wetzell
- Just checking out what you have here. Thus far, it is impressive!

- 44. John Atkinson
- This is an impressive site. However, my particular interest is Lebesgue integration, and none of the buttons under "the integral" seem to be active. Is the site still under construction? If so, when might these options be available? Thanks.

- 43. anoop
- Its an excellent attempt at teaching the very basic and the most fascinating field of mathematics. I, for one who is in Bombay India, have a very slow internet access. So it would be great if I may know of some ftp site which maintains this course, so that I can download it and install it for use by one and all here in the department of electrical engineering, Indian Institite of Technology, Bombay. Many thanks, Anoop Kulkarni

- 42. Einar Knudsen
- 41. HANS RINIKER
- I eagerly await a downloadable form of your very interesting course.

- 40. Craig Albert
- Neat! I only audited analysis, and I'd like to work through it again. I'll give it a try with your book. Craig Albert

- 39. Sharon Lesgold
- As a tutor for calculus students, I was curious to see what you had. It's a good refresher for me, at least. I'll be looking at it further. Thanks for posting this. --S. Lesgold

- 38. Kim Duk Sun
- Here site is very good!

- 37. Koray Atasoy
- But, where is Measure Theory, Lebesgue Integral etc.? I'm taking a Real Analysis course which is completely based on these topics. On the other hand, this is really a huge project...Congratulations

- 36. Derek W. Hein
- I will study real analysis next year. You'll hear more from me!!!!!!!

- 35. Kathy J. Doty
- Looks great!

- 34. Steve Leibel
- Looks great! I just took a quick look. I couldn't get it to work, exactly. I clicked on Lebesge Integration and nothing happened. I'm using Netscape 1.1b. I'll come back and spend some more time later. Acutally what I'd like is the second semester -- multi-variate calculus, inverse & implicit function theorems, elementary differential geometry, etc.

- 33. Steve Blum
- 32. Lino Notarantonio
- 31. A.Abhyanka
- An excellent idea. It will greatly help if a hard copy of the book -without the solutions could be made available. Solutions could be looked up interactively !

- 30. Lawrence Turyn
- 29. Koray Atasoy
- 28. Chris Kinsey
- Just started looking, so nothing yet.

- 27. Anne Booker
- 26. Tom Duquette
- 25. Mila
- I'm from Russia,Moscow.

- 24. Daniel Millar
- Making mathematics and physics books available online is very exciting idea. I hope others follow your lead and make such basic information as you have provided here available free of charge to the academic community in the future.

- 23. Henry Choy
- Snazzy, but rushed. Hook up some animation to it. I want to write notes on it. Is the book done yet? Some parts are missing.

- 22. Duane Mattern
- Looks real sweet. I LIKE IT!

- 21. JGuiciardi
- 20. H.R. Schaerer
- 19. Ole Martin Halck
- 18. Karen Covey
- 17. L.D. Simmons
- 16. Christopher J. Taylor
- 15. Anders Robertsson
- 14. Elliott Zimmermann
- 13. Anthony D. Thomas
- 12. Hagen Hagemann
- 11. Elliott Zimmermann
- 10. Anthony Suriano
- 9. Kerry Smith
- 8. Jean Paul Paradis
- 7. Lisa Netschert
- 6. Jennifer Nelson
- If the system printed out the material easily, it would be wonderful. Other than that, it's a great help.

- 5. Joy Matak
- 4. Peter Massung
- I feel it is very convenient as a resource, as it is better than most text books. (as long as the answers are present). Plus, it provides students with experience using the internet in which in the future will be helpful!

- 3. John Paul Kornak
- 2. Rachele Dembowski
- 1. Bert G. Wachsmuth

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