Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 2

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 $2.2.2$