P-adic Numbers is Hausdorff Topological Space

Theorem
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.

Then the topological space $\struct{\Q_p, \tau_p}$ is Hausdorff.

Proof
Let $d_p$ be the metric induced by the norm $\norm {\,\cdot\,}_p$.

By definition of the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$, $\tau_p$ is the topology induced by the metric $d_p$.

From Metric Space is Hausdorff, it follows that $\struct{\Q_p, \tau_p}$ is Hausdorff.