Metric Space is Hausdorff

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

Then $$M$$ is a Hausdorff space.

Proof
Let $$x, y \in A: x \ne y$$.

Then from Distinct Points in Metric Space have Disjoint Neighborhoods, there exist $\epsilon$-neighborhoods $$N_\epsilon \left({x}\right)$$ and $$N_\epsilon \left({y}\right)$$ which are disjoint metric spaces containing $$x$$ and $$y$$ respectively.

Hence the result by the definition of Hausdorff space.