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

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

Proof
$C > 1$.

By Characterizing Property of Supremum of Subset of Real Numbers:
 * $\exists m \in \N_{> 0}: \norm {m \cdot 1_R} > 1$

Let
 * $x = m \cdot 1_R$
 * $y = x^{-1}$

By Norm of Inverse:
 * $\norm y < 1$

By Sequence of Powers of Number less than One:
 * $\displaystyle \lim_{n \mathop \to \infty} \norm y^n = 0$

By Reciprocal of Null Sequence then:
 * $\displaystyle \lim_{n \mathop \to \infty} \frac 1 {\norm y^n} = +\infty$

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

{{eqn | r = \norm {x^n} | c = Norm axiom (N2) {Multiplicativity) }}

So:
 * $\displaystyle \lim_{n \mathop \to \infty} \norm {m^n \cdot 1_R} = +\infty$

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

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

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 $C = 1$.