## 2.4. The Real Number System

### Definition 2.4.2: Upper and Least Upper Bound

Let

**A**be an ordered set and**X**a subset of**A**. An element*b*is called an**upper bound**for the set**X**if every element in**X**is less than or equal to*b*. If such an upper bound exists, the set**X**is called**bounded above**.
Let **A** be an ordered set, and **X** a subset of **A**. An
element *b* in **A** is called a **least upper bound**__ __(or
**supremum**) for **X** if *b* is an upper bound for **X** and
there is no other upper bound *b'* for **X** that is less
than *b*. We write *b = sup(X)*.

By its definition, if a least upper bound exists, it is unique.