Completeness axiom of R

The completeness axiom (least upper bound property) for $\mathbb{R}$ states:

If $E\subseteq \mathbb{R}$ is nonempty and bounded above , then $E$ has a least upper bound in $\mathbb{R}$; that is, there exists $s\in\mathbb{R}$ such that

• $x\le s$ for all $x\in E$ (so $s$ is an upper bound), and
• if $u$ is any upper bound of $E$, then $s\le u$ (so $s$ is the least upper bound), and we write $s=\sup E$.

Completeness is the crucial axiom separating $\mathbb{R}$ from $\mathbb{Q}$ and is the source of many convergence and compactness results in real analysis.