Integers are Dense in P-adic Integers

Theorem
Let $p$ be a prime number.

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

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

Let $d_p$ be the metric induced by the norm $\norm {\,\cdot\,}_p$ restricted to the $p$-adic integers.

The integers $\Z$ are dense in the metric space $\struct{\Z_p, d_p}$.