Distance from Subset of Real Numbers to Supremum

Theorem
Let $S$ be a subset of the set of real numbers $\R$.

Let $x \in \R$ be a real number.

Let $\map d {x, S}$ be the distance between $x$ and $S$.

Let $S$ be bounded above such that $\xi = \sup S$.

Then:
 * $\map d {\xi, S} = 0$