Closed Balls Centered on P-adic Number is Countable/Open Balls/Lemma

Theorem
Let $p$ be a prime number.

Let $\epsilon \in \R_{\gt 0}$.

Then:
 * $\exists n \in \Z : p^{-n} < \epsilon \le p^{-n + 1}$

Proof
From Lemma 1:
 * $\exists m \in \Z : p^{-m} \le \epsilon < p^{-m + 1}$

Suppose $\epsilon \neq p^{-m}$.

Then:
 * $p^{-m} < \epsilon < p^{-m + 1}$

and the theorem is proved with $n = m$.

Now suppose $\epsilon = p^{-m}$. From Power Function on Integer between Zero and One is Strictly Decreasing:
 * $p^{-m - 1} < p^{-m}$

So:
 * $p^{-m - 1} < \epsilon \le p^{-m}$

and the theorem is proved with $n = m + 1$.