P-adic Numbers is Hausdorff Topological Space/Proof 2

Proof
Let $x, y \in \Q_p$ such that $x \ne y$.

By :
 * $r := \norm {x - y}_p > 0$

Then, for all $z\in\Q_p$ we have:

Therefore, the $r$-open balls $\map {B_r} x$ and $\map {B_r} y$ are disjoint.