Point in Finite Hausdorff Space is Isolated

Theorem
Let $$T = \left({A, \vartheta}\right)$$ be a Hausdorff 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 $$X = \left\{{x_1, x_2, \ldots, x_n}\right\}$$.

Let $$x_k \in X$$.

From the definition of Hausdorff space, $$\forall x_i, x_j \in X: x_i \ne x_j: \exists U, V \in \vartheta: x_i \in U, x_j \in V: U \cap V = \varnothing$$.

So there is a neighborhood of $$x_k$$ containing only $$x_k$$, so by definition, $$x_k$$ is an isolated point.

Hence the result.