Existence of Sequence in Subset of Metric Space whose Limit is Infimum

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

Let $a \in A$.

Let $S \subseteq A$ be a non-empty subset of $A$.

Then there exists a sequence $\left\langle{a_n}\right\rangle$ of points of $S$ such that:
 * $\displaystyle \lim_{n \mathop \to \infty} d \left({a, a_n}\right) = d \left({a, S}\right)$

Proof
From Existence of Sequence in Set of Real Numbers whose Limit is Infimum:
 * $\displaystyle \lim_{n \mathop \to \infty} a_n = b$

where $b$ is an infimum of $d \left({a, a_n}\right)$.

Hence the result by definition of distance to a subset of a metric space.