Distance from Subset to Element

Theorem
Let $\struct {M, d}$ be a metric space.

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

Let $s \in S$.

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

where $\map d {s, S}$ denotes the distance between $s$ and $S$.

Proof
By Distance between Element and Subset is Nonnegative:


 * $\map d {s, S} \ge 0$

Also, because:


 * $\map d {s, S} = 0$

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


 * $\map d {s, S} \le 0$

Hence the result.