Let $A \subset \mathbb{R}$ be nonempty and bounded above, and let $s \in \mathbb{R}$. Then $s = \sup A$ if and only if $s$ is an upper bound for $A$ and, for every $\varepsilon > 0$, there exists $a \in A$ such that
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\begin{align*}
s - \varepsilon < a \leq s.
\end{align*}