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.

$\blacksquare$