All metric spaces are Hausdorff spaces

up:: Metric Topology

Let $(M,d)$ be a Metric Space, and $x,y\in M$.

Then let their respective Open Balls (Metric Spaces) $B_r(x),B_r(y)$ both with radius $r=\frac{d(x,y)}{2}$.

Note that $B_r(x)\cap B_r(y)=\varnothing$.
Proof by absurd:
Let $z\in B_r(x)\cap B_r(y)$. Then $d(x,z)<r$ and $d(z,y)<r$.
Thus, $d(x,y)\le d(x,z)+d(z,y)<2r=d(x,y)$, which is absurd.

Thus, since we have found open sets which separate these points, then this space is a Hausdorff Space.