Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition

Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.

Then:
 * $\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1 \implies \norm {\,\cdot\,}$ is non-archimedean

where:
 * $n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{n \text { times} }$

Proof
Let:
 * $\forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$

Let $x, y \in R$.

Let $y = 0_R$ where $0_R$ is the zero of $R$.

Then $\norm {x + y} = \norm x = \max \set {\norm x, 0} = \max \set {\norm x, \norm y}$

Lemma 1
Hence to complete the proof it is sufficient to prove:
 * $\forall x \in R: \norm {x + 1_R} \le \max \set {\norm x, 1}$

For $n \in \N$:

Lemma 2
Hence

Taking $n$th roots yields:
 * $\norm {x + 1_R} \le \paren {n + 1}^{1/n} \max \set {\norm x, 1}$

Lemma 3
By the Multiple Rule for Real Sequences:
 * $\ds \lim_{n \mathop \to \infty} \paren {n + 1}^{1/n} \max \set {\norm x, 1} = \max \set {\norm x, 1}$

By Inequality Rule for Real Sequences:
 * $\norm {x + 1_R} \le \max \set {\norm x, 1}$

The result follows.