Supremum of Bounded Above Set of Reals is in Closure

Theorem
Let $\R$ be the real numbers as a metric space under the Euclidean metric.

Let $H \subseteq \R$ be a bounded above subset of $\R$.

Let $u = \sup \left({H}\right)$ be the supremum of $H$.

Then:
 * $u \in \operatorname{cl}\left({H}\right)$

where $\operatorname{cl}\left({H}\right)$ denotes the closure of $H$ in $\R$.