Let $\displaystyle S$ be a bounded non-empty subset of $\displaystyle \mathbb{R}$, and $\displaystyle \bar m = sup S$.

Prove there is a sequence $\displaystyle {a_{n}}$ such that $\displaystyle a_{n} \in S$ for all $\displaystyle n$,

and $\displaystyle a_{n} \rightarrow \bar m$