Uniqueness of the Supremum and of the Infimum

Let $A$ be a set equipped with a total order $\le$, and let $X\subseteq A$. Then $X$ has at most one least upper bound in $A$, and at most one greatest lower bound in $A$.

Accordingly, when a least upper bound of $X$ exists it is denoted $\sup X$, and when a greatest lower bound of $X$ exists it is denoted $\inf X$.