Integers with Metric Induced by P-adic Valuation

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

Let $d: \Z^2 \to \R_{\ge 0}$ be the mapping defined as:


 * $\forall x, y \in \Z: \map d {x, y} = \begin {cases} 0 & : x = y \\ \dfrac 1 r & : x - y = p^{r - 1} k: r, k \in \Z, p \nmid k \end {cases}$

Then $d$ is a metric on $\Z$.