Completeness Axiom

Completeness axiom (least upper bound property): If $A \subseteq \mathbb{R}$ is nonempty and bounded above , then $A$ has a supremum in $\mathbb{R}$.

This axiom distinguishes $\mathbb{R}$ from other ordered fields and underlies many foundational results, including the monotone sequence convergence theorem and the equivalences collected in completeness equivalences .