Completeness

Completeness is a key property of the set of real numbers that essentially explains the continuity of the real number line. The corresponding axiom of completeness states that:

Every non-empty set of real numbers that is bounded above has a least upper bound.¹

Under the axiom’s statement, we assume it holds true without proof. However, it’s possible to prove it as a theorem when we rigorously construct the real numbers.

Some definitions while we’re here:

• A set $A\subset \mathbb{R}$ is bounded above if there exists a number $b\in \mathbb{R}$ such that $a\le b\forall a\in A$. $b$ is an upper bound.
• Similarly, the set $A$ is bounded below if there is a lower bound $l\in \mathbb{R}$ such that $l\le a\forall a\in A$.
• The least upper bound (also the supremum) for a set $A\subset \mathbb{R}$ is a real number $s=\text{lub}A=\text{sup}A$ if the following properties hold, i.e., it is the smallest upper bound for the set:
  • $s$ is an upper bound for $A$, and
  • If $b$ is any upper bound for $A$, then $s\le b$.
• A real number $a_0$ is a maximum of a set $A$ if $a_0\in A$ and $a_0\ge a\forall a\in A$.
• Similarly, $a_1$ is a minimum of $A$ if $a_1\in A$ and $a_1\le a\forall a\in A$.

Implications

Footnotes

1. From Understanding Analysis, by Stephen Abbott. ↩