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

Proof
The integers $\Z$ are a subring of the $p$-adic integers $Z_p$ by Integers form Subring of P-adic Integers.

The $p$-adic integers is closed in the $p$-adic metric by Set of P-adic Integers is Clopen in P-adic Numbers.

By Closure of Subset of Closed Set of Metric Space is Subset then the closure of $\Z$ is contained in $\Z_p$:
 * $\map {cl} \Z \subseteq \Z_p$