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 if and only if:

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

where:

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

Proof

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

By Corollary 2:

$\sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } > 1 \iff \sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } = +\infty$

$\blacksquare$