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 that is bounded above has a supremum in $\mathbb{R}$.

The Completeness Property is also often called the "Least Upper Bound Property".

The completeness property above is a crucial axiom. A similar theorem regarding nonempty subsets $S$ of the real numbers that are bounded below exists and is proven below.

Theorem 1: Every nonempty subset $S$ of the real numbers that is bounded below has an infimum in $\mathbb{R}$.
  • Proof: Let $S$ be a nonempty subset of real numbers that is bounded below. Then there exists a lower bound $m \in \mathbb{R}$ such that $m \leq s$ for every $s \in S$.
  • Define the set $T$ by:
(1)
\begin{align} \quad T = \{ -s : s \in S \} \end{align}
  • Then observe that $-s \leq -m$ for every $-s \in T$. So $T$ is a nonempty subset of real numbers that is bounded above. By the completeness property of the real numbers we have that $T$ has a supremum in $\mathbb{R}$. Let $x = \sup T$. Then $-s \leq x$ for every $-s \in T$ and if $y$ is any other upper bound of $T$ then $x \leq y$.
  • So observe that $x \leq s$ for every $s \in S$ and if $y$ is any other lower bound of $S$ then $y \leq x$. So $\inf S = -x$. In other words, $S$ has an infimum in $\mathbb{R}$. $\blacksquare$
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