Valuation Ring of Non-Archimedean Division Ring is Clopen

Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$.

Let $\OO$ be valuation ring induced by $\norm{\,\cdot\,}$.

Then $\OO$ is a both open and closed in the metric induced by $\norm{\,\cdot\,}$.

Proof
The valuation ring $\OO$ Is the open ball $\map {B_1^-} {0_R}$ by definition.

By Open Balls of Non-Archimedean Division Rings are Clopen then $\OO$ is both open and closed in the metric induced by $\norm {\,\cdot\,}$.