Uniqueness of Supremum and Infimum

Uniqueness of supremum/infimum: Let $A \subseteq \mathbb{R}$ be nonempty.

• If $s$ and $t$ are both suprema of $A$, then $s=t$.
• If $u$ and $v$ are both infima of $A$, then $u=v$.

This guarantees that the notation $\sup A$ and $\inf A$ is unambiguous whenever these numbers exist, which is ensured under the hypotheses of the completeness axiom (for $\sup$) and its dual (for $\inf$).