Characterisation of Non-Archimedean Division Ring Norms/Corollary 2

Theorem

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

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

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

Then $\norm {\,\cdot\,}$ is non-Archimedean and $C = 1$.

Proof

Aiming for a contradiction, suppose $C > 1$.

$\exists m \in \N_{> 0}: \norm {m \cdot 1_R} > 1$

Let

$x = m \cdot 1_R$
$y = x^{-1}$
$\norm y < 1$
$\ds \lim_{n \mathop \to \infty} \norm y^n = 0$

By Reciprocal of Null Sequence then:

$\ds \lim_{n \mathop \to \infty} \frac 1 {\norm y^n} = +\infty$

For all $n \in \N_{> 0}$:

 $\ds \dfrac 1 {\norm y^n}$ $=$ $\ds \norm {y^{-1} }^n$ Norm of Inverse $\ds$ $=$ $\ds \norm x^n$ Definition of $y$ $\ds$ $=$ $\ds \norm {x^n}$ Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity $\ds$ $=$ $\ds \norm {\paren {m \cdot 1_R}^n}$ Definition of $x$ $\ds$ $=$ $\ds \norm {m^n \cdot 1_R}$

So:

$\ds \lim_{n \mathop \to \infty} \norm {m^n \cdot 1_R} = +\infty$

Hence:

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

This contradicts the assumption that $C < +\infty$.

$\Box$

It follows that $C \le 1$.

Then:

$\forall n \in \N_{>0}: \norm{n \cdot 1_R} \le 1$

By Characterisation of Non-Archimedean Division Ring Norms, $\norm{\,\cdot\,}$ is non-Archimedean.

By Corollary 1:

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

So $C = 1$.

$\blacksquare$