The Completeness Property of The Real Numbers

# The Completeness Property of The Real Numbers

We will now look at yet again another crucially important property of the real numbers which will allow us to call the set of $\mathbb{R}$ numbers under the operations of addition and multiplication a **complete ordered field**. This property will ensure that there is no "gaps" in the real number line, that is the real number line is continuous. The property is as follows.

The Completeness Property of The Real Numbers: Every nonempty subset $S$ of the real numbers $\mathbb{R}$ that is bounded above has a supremum in $\mathbb{R}$. Similarly, every nonempty subset $S$ of the real numbers $\mathbb{R}$ has is bounded below has an infimum in $\mathbb{R}$. |