Finite Subset of Metric Space has No Limit Points

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

Follows directly from:

• All Points in Finite Metric Space are Isolated
• Point is Isolated Iff Not a Limit Point.

$\blacksquare$