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 :
 * $\sup \set {\norm{n \cdot 1_R}: n \in \N_{\gt 0}} = +\infty$.

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 Archimedean $\,\,\iff \sup \set {\norm{n \cdot 1_R}: n \in \N_{\gt 0}} \gt 1$

By Corollary 2 then:
 * $\,\,\sup \set {\norm{n \cdot 1_R}: n \in \N_{\gt 0}} \gt 1 \iff \sup \set {\norm{n \cdot 1_R}: n \in \N_{\gt 0}} = +\infty$