Characterisation of Non-Archimedean Division Ring Norms/Corollary 2

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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 $C > 1$.


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:

$\ds \lim_{n \mathop \to \infty} \norm y^n = 0$


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 $C = 1$.

$\blacksquare$


Sources

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