Restricted P-adic Metric is Metric

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

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


 * $\forall x, y \in \Z: \map {d^\Z_p} {x, y} = \norm {x - y}_p$

where $\norm {x - y}_p$ denotes the $p$-adic norm.

Then $d^\Z_p$ is a metric.

Proof
From $p$-adic Metric is Metric, the $p$-adic metric on $\Q$:
 * $\forall x, y \in \Q: \map {d_p} {x, y} = \norm {x - y}_p$

forms a metric space $\struct {\Q, d_p}$.

The mapping:
 * $\forall x, y \in \Z: \map {d^\Z_p} {x, y} = \norm {x - y}_p$

is the restriction of $d_p$ to the integers.

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

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