Let $S \subseteq \mathbb{R}$ be nonempty and bounded above. A number $s$ is the supremum of $S$ if: (i) $s$ is an upper bound of $S$ (see \ref{def:upper-bound-rudin-b}); and (ii) for every $\varepsilon>0$ there exists $x\in S$ with $s-\varepsilon < x$. We write $s=\sup S$.