P-adic Integers is Metric Completion of Integers

Theorem
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Then the $p$-adic integers $\Z_p$ is the metric completion of the integers $\Z$ with the $p$-adic metric.