Metric Space is Hausdorff

Theorem
Let $M = \struct {A, d}$ 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 Open Balls, there exist open $\epsilon$-balls $\map {B_\epsilon} x$ and $\map {B_\epsilon} Y$ which are disjoint open sets containing $x$ and $y$ respectively.

Hence the result by the definition of Hausdorff space.