Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 2
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Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.
Let $x \in R$.
Let $n \in \N$.
Then for all $i$, $0 \le i \le n$:
- $\norm x^i \le \max \set {\norm x^n , 1}$
Proof
If $\norm x > 1$ then for all $i$, $0 \le i \le n$:
- $\norm x^i \le \norm x^n \le \max \set {\norm x^n, 1}$
If $\norm x \le 1$ then for all $i$, $0 \le i \le n$:
- $\norm x^i \le 1 \le \max \set {\norm x^n, 1}$
In either case for all $i$, $0 \le i \le n$:
- $\norm x^i \le \max \set {\norm x^n , 1}$
$\blacksquare$
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.2$ Normed Fields, Proposition $1.14$
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.2$ Basic Properties, Theorem $2.2.2$