Proof of Uniqueness of the Supremum and of the Infimum

Suppose $s$ and $s'$ are both least upper bounds of $X$ in the sense of Upper Bound and Least Upper Bound. In particular each is an upper bound for $X$. Since $s$ is a least upper bound and $s'$ is an upper bound, $s\le s'$; since $s'$ is a least upper bound and $s$ is an upper bound, $s'\le s$. By the antisymmetry axiom of a total order, $s=s'$.

Suppose $m$ and $m'$ are both greatest lower bounds of $X$ in the sense of Lower Bound and Greatest Lower Bound in a Totally Ordered Set. In particular each is a lower bound for $X$. Since $m'$ is a greatest lower bound and $m$ is a lower bound, $m\le m'$; since $m$ is a greatest lower bound and $m'$ is a lower bound, $m'\le m$. By antisymmetry, $m=m'$.

Thus $X$ has at most one least upper bound and at most one greatest lower bound, so the notations $\sup X$ and $\inf X$ are unambiguous whenever the corresponding bound exists.