# Distance from Subset of Real Numbers to Element/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 $S$.

Then:

$x \in S \implies \map d {x, S} = 0$

## 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 Element.

$\blacksquare$