Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition

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

Then:
 * $\norm{\,\cdot\,}$ is non-archimedean $\implies \forall n \in \N_{>0}: \norm{n \cdot 1_R} \le 1$.

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

Proof
Let $\norm{\,\cdot\,}$ be non-archimedean.

Then by the definition of a non-archimedean norm, for $n \in \N$,