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A real number $s$ is the supremum of a nonempty set $A\subset \mathbb{R}$ if every $a\in A$ satisfies $a\le s$ and no smaller real number is an upper bound. Equivalently, for every $\eta >0$ there is $a\in A$ with $s-\eta <a\le s$. A greatest element, or maximum, additionally belongs to $A$.

Let $s=\sup A$ and suppose $s\notin A$. Fix $\epsilon >0$. The approximation property gives at least one element of $A$ in $(s-\epsilon ,s)$. If there were only finitely many, they would have a largest value $m<s$. Every other element of $A$ is at most $s-\epsilon$, so $\max \{m,s-\epsilon \}<s$ would be an upper bound for the whole set. This contradicts the definition of the supremum. Therefore there are infinitely many such elements, the accumulation below an unattained supremum.

For a counterexample when the supremum is attained, take $A=\{0\}$. Its supremum and greatest element are both zero, while every interval $(-\epsilon ,0)$ contains no elements of $A$.