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 $C > 1$.
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
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.2$ Basic Properties: Problem $41$