$B(a, p^n) = \bar{B}(a, p^{n-1})$ in $\mathbb{Q_p}$

Question

In an article about $p$-adic numbers I have read that

$B(a, p^n) = \{x \in \mathbb{Q}_p \mid \left\| x - a \right\|_p < p^n \} = \{x \in \mathbb{Q}_p \mid \left\| x - a \right\|_p \leq p^{n-1}\} = \bar{B}(a, p^{n-1})$, where $n \in \mathbb{Z}$ and sadly, I don´t understand this. Can someone please give me an explanation ?

Thanks for your help.

Answer 1

In $\Bbb Q_p$ the only possible nonzero distances between elements are $p^m$ for $m\in\Bbb Z$. Therefore if $\|x\|_p<p^n$ ($n\in\Bbb Z$) then $\|x\|_p\le p^{n-1}$.