Supremum Approximation Lemma

Supremum approximation lemma: Let $A \subseteq \mathbb{R}$ be nonempty and bounded above, and let $s=\sup A$ (see supremum). Then for every $\varepsilon>0$ there exists $a\in A$ such that $s-\varepsilon < a \le s$.

Dually, if $A$ is nonempty and bounded below with $m=\inf A$, then for every $\varepsilon>0$ there exists $a\in A$ such that $m \le a < m+\varepsilon$.

This lemma is the standard way to pass between statements involving $\sup A$ or $\inf A$ and $\varepsilon$-style inequalities used in limits and estimates.