Characterisation of Non-Archimedean Division Ring Norms/Corollary 3

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

$\norm{\,\cdot\,}$ is archimedean :
 * $\sup \set {\norm{n \cdot 1_R}: n \in \Z} = +\infty$.

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