Characterisation of Non-Archimedean Division Ring Norms/Corollary 5

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

If $\norm{\,\cdot\,}$ is non-Archimedean then:


 * $\sup \set {\norm{n \cdot 1_R}: n \in \Z} = 1$.

where $n \cdot 1_R = \begin{cases} \underbrace {1_R + 1_R + \dots + 1_R}_{n \, times} &\mbox{if } n \gt 0 \\ 0 &\mbox{if } n = 0 \\ \\ -\underbrace {\paren {1_R + 1_R + \dots + 1_R}}_{-n \, times} &\mbox{if } n \lt 0 \\ \end{cases}$

Proof
By Corollary 1 of Characterisation of Non-Archimedean Division Ring Norms then:
 * $\sup \set {\norm{n \cdot 1_R}: n \in \N_{\gt 0}} = 1$.

By Norm Axiom (N1) (Positive Definiteness) then:
 * $\norm{0 \cdot 1_R} = 0 \le 1$

Let $n \lt 0$ then:

The result follows.