Every Metric Spaces is Hausdorff

We will prove this by proving a more general and constructive result which will trivially imply the Hausforff property.

Proof

We prove this by the contrapositive, that is assume that

Therefore

Now, summing these inequalities and applying the triangle inequality we have that

Therefore we have that $d(x,y)<r_x+r_y$, and therefore by the contrapositive

Now we prove the Hausdorff property.

Proof

For any $x,y\in X$ with $x\ne y$ let $r_x=r_y=\frac{d(x,y)}{2}$. As such, $r_x+r_y=\frac{d(x,y)}{2}\le \frac{d(x,y)}{2}$ as required to apply our lemma. Therefore