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 subspace metric of 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.

Hence $\Z \subseteq \Z_p$

The $p$-adic integers $\Z_p$ 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 $\Z_p$:
 * $\map {\operatorname{cl}} \Z \subseteq \Z_p$

By P-adic Integer is Limit of Unique Coherent Sequence of Integers then:
 * for all $x \in \Z_p$ there exists a sequence $\sequence {x_n}$ in $\Z$ such that $\displaystyle \lim_{n \to \infty} x_n = x$

By Closure of Subset of Metric Space by Convergent Sequence then:
 * $\Z_p \subseteq \map {\operatorname{cl}} \Z$

Hence $\map {\operatorname{cl}} \Z = \Z_p$

By Metric Subspace Induces Subspace Topology then the topology induced by the $p$-adic metric on $\Z_p$ is the subspace topology.

By Closure in Subspace then $\map {\operatorname{cl}} \Z = \Z_p$ in the subspace metric on $\Z_p$.

It follows that $\Z_p$ with the subspace metric is the metric completion of $\Z$.