Point in Finite Metric Space is Isolated

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

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

Then all points of $X$ are isolated.

Proof
As $X$ is finite, its elements can be placed in one-to-one correspondence with the elements of $\N^*_n$ for some $n \in \N$.

So let $X = \left\{{x_1, x_2, \ldots, x_n}\right\}$.

Now let $\epsilon = \min \left\{{\forall i, j \in \N^*_n: i \ne j: d \left({x_i, x_j}\right)}\right\}$.

That is, $\epsilon$ is the minimum distance between any two elements of $X$.

We have that $\epsilon > 0$ from the definition of metric.

Let $N_{\epsilon} \left({x_k}\right)$ be the $\epsilon$-neighborhood of any $x_k \in X$.

By the method of construction of $\epsilon$, it is clear that $N_{\epsilon} \left({x_k}\right) = \left\{{x_k}\right\}$.

Hence $x_k$ is an isolated point.

Alternative Proof
Alternatively, it can be noted that as a Metric Space is Hausdorff, we can apply the result All Points in Finite Hausdorff Space are Isolated‎.