P-adic Metric is Metric

Theorem
Let $p \in \N$ be a prime.

Let $\left\Vert{\cdot}\right\Vert_p: \Q \to \R_{\ge 0}$ be the $p$-adic norm on $\Q$.

Let $d_p$ be the $p$-adic metric on $\Q$:


 * $\forall x, y \in \Q: d \left({x, y}\right) = \left\Vert{x - y}\right\Vert_p$

Then $d_p$ is a metric.

Proof
The $p$-adic metric on $\Q$ is defined as the metric induced by the $p$-adic norm on $\Q$.

It follows from Metric Induced by Norm is Metric that $d_p$ is a metric.