P-adic Metric is Metric

Theorem

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

Let $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ be the $p$-adic norm on $\Q$.

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

$\forall x, y \in \Q: \map {d_p} {x, y} = \norm{x - y}_p$

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.

$\blacksquare$