Distance from Subset to Element
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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.
$\blacksquare$