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 $\norm{x} \gt 1$ 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 $i$, $0 \le i \le n$, then:

$\norm {x}^i \le \max \set {\norm {x}^n , 1}$

$\blacksquare$

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