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 $n \cdot 1_R = \begin{cases} \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} } & : n > 0 \\ 0 & : n = 0 \\ \\ -\underbrace {\paren {1_R + 1_R + \dots + 1_R} }_{\text {$-n$ times} } & : n < 0 \\ \end{cases}$

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

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

Let $n < 0$ then:

The result follows.