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

From ProofWiki
Jump to navigation Jump to search

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