Topological Properties of Non-Archimedean Division Rings/Closed 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 $r \in \R_{\gt 0}$.

Let ${B_r}^- \paren{x}$ be the closed $r$-ball of $x$ in $\struct {R,d}$

Then:
 * The closed $r$-ball of $x$, ${B_r}^- \paren x$, is both open and closed.