Distance from Subset to Element

Theorem
Let $\left({M, d}\right)$ be a metric space.

Let $S \subseteq M$ be a subset of $M$.

Let $s \in S$.

Then $d \left({s, S}\right) = 0$, where $d \left({s, S}\right)$ denotes the distance between $s$ and $S$.

Proof
By Distance between Element and Subset is Nonnegative:


 * $d \left({s, S}\right) \ge 0$

Also, since:


 * $d \left({s, s}\right) = 0$

and $s \in S$, it follows that:


 * $d \left({s, S}\right) \le 0$

Hence the result.