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

Hence:
 * $p^{-v} > 1 = p^0$.

From Real Power Function on Base Greater than One is Strictly Increasing:
 * $-v > 0$.

Let $n = -v$.

Then:

The result follows.