Supremum approximation lemma

Supremum approximation lemma: Let $E\subseteq\mathbb{R}$ be nonempty and bounded above , and let $s=\sup E$. Then:

• for every $\varepsilon>0$ there exists $x\in E$ such that $s-\varepsilon < x \le s,$
• equivalently, there exists a sequence $(x_n)$ in $E$ such that $x_n\to s$.

This lemma is used constantly to turn the abstract existence of $\sup E$ into a usable approximation statement (often in $\varepsilon$–arguments).

Examples:

• If $E=(0,1)$, then $\sup E=1$, and one may take $x_n=1-\frac{1}{n}$.
• If $E=\{x\in\mathbb{R}:x^2<2\}$, then $\sup E=\sqrt{2}$, and the lemma guarantees a sequence $x_n\uparrow \sqrt{2}$ with $x_n^2<2$.