P-adic Number times P-adic Norm is P-adic Unit

Theorem
Let $p$ be a prime number.

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

Let $\Z_p^\times$ be the $p$-adic units.

Then:
 * $\forall a \in \Q_p: \exists n \in \Z: p^n a \in \Z_p^\times$.

Proof
Let $a \in \Q_p$.

From P-adic Norm of p-adic Number is Power of p, there exists $v \in \Z$ such that $\norm a_p = p^{-v}$.

Let $n = -v$.

Then:

The result follows.