Non-Archimedean Norm iff Non-Archimedean Metric/Sufficient Condition

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

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

Let $d$ be non-Archimedean.

Then:
 * $\norm {\,\cdot\,}$ is a non-Archimedean norm.

Proof
Let $x, y \in R$.

Hence $\norm {\,\cdot\,}$ is non-Archimedean.