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

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:

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

$\blacksquare$

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