Lower Bound and Greatest Lower Bound in a Totally Ordered Set

Let $A$ be a set equipped with a total order $\le$, and let $X\subseteq A$. An element $\ell\in A$ is a lower bound for $X$ if $\ell\le x$ for every $x\in X$. If such an $\ell$ exists, then $X$ is bounded below. An element $m\in A$ is a greatest lower bound, or infimum, of $X$ if $m$ is a lower bound for $X$ and $\ell\le m$ for every lower bound $\ell$ of $X$.