Finite Subset of Metric Space has no Limit Points

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Theorem

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

Let $X \subseteq A$ such that $X$ is finite.


Then $X$ has no limit points.


Proof

Let $x \in X$.

From Point in Finite Metric Space is Isolated, $x$ is an isolated point.

The result follows by definition of isolated point:

$x$ is an isolated point if and only if $x$ is not a limit point.

$\blacksquare$


Sources