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 $n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{n \, times}$