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

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

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