Characterisation of Non-Archimedean Division Ring Norms/Corollary 3

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Theorem

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


$\norm {\,\cdot\,}$ is Archimedean if and only if:

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


where:

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


Proof

By Characterisation of Non-Archimedean Division Ring Norms:

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


By Corollary 2:

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

$\blacksquare$


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