## 2.4. The Real Number System

### Definition: Lower and Greatest Lower Bound

Let

**A**be an ordered set and**X**a subset of**A**. An element*b*is called a**lower bound**for the set**X**if every element in**X**is greater than or equal to*b*. If such a lower bound exists, the set**X**is called**bounded below**.
Let **A** be an ordered set, and **X** a subset of **A**. An
element *b* in **A** is called a **greatest lower bound**__ __
(or **infimum**) for **X** if *b* is a lower bound for **X**
and there is no other lower bound *b'* for **X** that is greater
than *b*. We write *b = inf( X)*.

By its definition, if a greatest lower bound exists, it is unique.