Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.2

Theorem

Let $R$ be a division ring.

Let $\norm {\, \cdot \,}_1: R \to \R_{\ge 0}$ and $\norm {\, \cdot \,}_2: R \to \R_{\ge 0}$ be norms on $R$.

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:

$\forall y \in R:\norm y_1 \lt 1 \iff \norm y_2 \lt 1$

Then:

$\forall y \in R:\norm y_1 = 1 \iff \norm y_2 = 1$

By assumption:

$\forall y \in R:\norm y_1 \ge 1 \iff \norm y_2 \ge 1$

$\forall y \in R:\norm y_1 \le 1 \iff \norm y_2 \le 1$

Hence $\forall y \in R$:

$\ds \norm y_1 = 1$

$\leadstoandfrom$

$\ds \norm y_1 \le 1, \norm y_1 \ge 1$

$\ds $

$\leadstoandfrom$

$\ds \norm y_2 \le 1, \norm y_2 \ge 1$

Lemma 1 and by assumption

$\ds $

$\leadstoandfrom$

$\ds \norm y_2 = 1$

$\blacksquare$