Upper Bound and Least Upper Bound

Let $A$ be a set equipped with a total order $\le$, and let $X\subseteq A$. An element $b\in A$ is an upper bound for $X$ if $x\le b$ for every $x\in X$. If such a $b$ exists, then $X$ is bounded above. An element $s\in A$ is a least upper bound, or supremum, of $X$ if $s$ is an upper bound for $X$ and $s\le b$ for every upper bound $b$ of $X$.