Uniqueness of supremum and infimum

Let $E\subseteq\mathbb{R}$.

A real number $s$ is the supremum of $E$ if:

• $x\le s$ for all $x\in E$ (so $s$ is an upper bound ), and
• for every $u\in\mathbb{R}$, if $x\le u$ for all $x\in E$ then $s\le u$ (so $s$ is the least upper bound).

Uniqueness of supremum: If $s$ and $t$ are both $\sup E$, then $s=t$.

Uniqueness of infimum: If $s$ and $t$ are both $\inf E$, then $s=t$.

Uniqueness is needed to treat $\sup E$ and $\inf E$ as well-defined numbers rather than as “choices.”

Proof sketch: If $s$ and $t$ are both suprema, then since $t$ is an upper bound, the “least upper bound” property of $s$ gives $s\le t$. Symmetrically $t\le s$. Hence $s=t$. The infimum case is identical (or reduce to supremum via $\inf E=-\sup(-E)$).