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 $\mathcal O$ be valuation ring induced by $\norm{\,\cdot\,}$.

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

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

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