Open and closed balls in $\mathbb{Q}_p$

Question

Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. The open and closed ball with center $a$ and radius $r$ in $\mathbb{Q}_p$ are defined as follows:

$B(a, r) = \{x \in \mathbb{Q}_p \mid \left|x - a \right|_p < r\}$ and $\bar{B}(a, r) = \{x \in \mathbb{Q}_p \mid \left|x - a \right|_p \leq r \}$.

How can I prove that for $0 < r \in \mathbb{R} \backslash \{p^z \mid z \in \mathbb{Z}\}$ one has $\bar{B}(a,r) = B(a,r)$ ? Can anyone give me some advice ?

Thanks in advance.

Answer 1

To get this off the "unanswered" list, this [slightly corrected] comment by user Hw Chu (Sep 29, 2018) answers it:

The value of $|x|_p$ can only be of the form $p^a$, where $a \in \mathbb Z$. Saying $|x|_3<3^{1.1}$ or $|x|_3 \le 3^{1.1}$ is really the same as saying $|x|_3 \le 3^1$.