P-adic Number times Integer Power of p is P-adic Integer

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.

Then:
 * $\forall a \in \Q_p: \exists n \in \N: p^n a \in \Z_p$

Proof
Let $a \in \Q_p$.

Case: $\norm a_p \le 1$
Let $\norm a_p \le 1$.

By definition of the $p$-adic integers:
 * $a \in \Z_p$.

Hence:
 * $p^0 a \in \Z_p$.

Case: $\norm a_p > 1$
Let $\norm a_p > 1$.

From P-adic Number is Integer Power of p times P-adic Unit, there exists $n \in \Z$ such that $p^n a$ is a $p$-adic unit.

Then: