Open and Closed Balls in P-adic Numbers are Compact Subspaces

Theorem
Let $p$ be a prime number.

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

Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.

Let $\Z_p$ be the $p$-adic integers.

Then the topological subspace $\Z_p$ is compact.