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

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Theorem

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

Let $a \in A$.

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


Then there exists a sequence $\sequence {a_n}$ of points of $S$ such that:

$\displaystyle \lim_{n \mathop \to \infty} \map d {a, a_n} = \map d {a, S}$


Proof

From Existence of Sequence in Set of Real Numbers whose Limit is Infimum:

$\displaystyle \lim_{n \mathop \to \infty} \map d {a, a_n} = b$

where $b$ is an infimum of $\map d {a, a_n}$.

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

$\blacksquare$


Sources