Characterisation of Non-Archimedean Division Ring Norms

Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.

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


 * $\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$

where:
 * $n \cdot 1_R = \underbrace {1_R + 1_R + \dotsb + 1_R}_{\text {$n$ times} }$