Distance from Subset of Real Numbers to Supremum/Proof 2

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:

$\map d {\xi, S} = 0$

Proof

Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.

The result is then seen to be an example of Distance from Subset to Supremum.

$\blacksquare$