Restricted P-adic Metric is Metric

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

Let $d^\Z_p$ be the $p$-adic metric on $\Z$:


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

where $\left\Vert{x - y}\right\Vert_p$ denotes the $p$-adic norm.

Then $d^\Z_p$ is a metric.

Proof
From $p$-adic Metric is Metric, the $p$-adic metric on $\Q$:
 * $\forall x, y \in \Q: d_p \left({x, y}\right) = \left\Vert{x - y}\right\Vert_p$

forms a metric space $\left({\Q, d_p}\right)$.

The mapping:
 * $\forall x, y \in \Z: d^\Z_p \left({x, y}\right) = \left\Vert{x - y}\right\Vert_p$

is the restriction of $d_p$ to the integers.

Hence the $p$-adic metric on $\Z$ is a metric subspace $\left({\Z, d^\Z_p}\right)$ of $\left({\Q, d_p}\right)$.

The result follows from Subspace of Metric Space is Metric Space.