Integers are Arbitrarily Close to 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 $x \in \Z_p$.

Then for $n \in \N$ there exists unique $\alpha \in \Z$:
 * $0 \le \alpha \le p^n - 1$
 * $\norm { x -\alpha}_p \le p^{-n}$