## 2.4. The Real Number System

### Examples 2.4.3(a):

Consider the set S of all rational numbers strictly between 0 and 1. Then
this set has many upper bounds, but only one least upper bound. That supremum
does not have to be part of the original set.

An upper bound for the set **S**is any number that is greater than or equal to any number in the set

**S**. Five different upper bounds for

**S**are, for example:

- 1, 10, 100, 42, and e (Euler's number)

**S**.

To find the least upper bound for **S** we need to find a number *x*
such that

*x*is an upper bound for**S**- there is no other upper bound lower than
*x*

If we include 0 and 1 in the original set S, then the least upper bound is
again 1. This time the unique supremum is part of the set **S**, which
may or may not happen in general.