Characterisation of Non-Archimedean Division Ring Norms/Corollary 1

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

$\norm{\,\cdot\,}$ is non-Archimedean :
 * $\sup \set {\norm{n \cdot 1_R}: n \in \Z} = 1$.

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

Proof
By Characterisation of Non-Archimedean Division Ring Norms then:
 * $\norm{\,\cdot\,}$ is non-Archimedean $\,\,\sup \set {\norm{n \cdot 1_R}: n \in \Z} \le 1$.

By norm of unity then:
 * $\norm {1_R} = 1$

The result follows.