Topological Properties of Non-Archimedean Division Rings/Open Balls are Clopen

Theorem
Let $\struct {R, \norm{\,\cdot\,}}$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$,

Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$.

Let $x \in R$.

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

For $x \in R$ and $r \in \R_{>0}$ let:
 * $B_\epsilon \paren{a}$ denote the open $r$-ball of $x$ in $\struct {R,d}$

Then:
 * The set $B_r \paren x$ is both open and closed.