Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm/Lemma 2/Lemma 2.1

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:

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

Then:

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

For $y \in R$ then:

$\ds \norm y_1 > 1$

$\leadstoandfrom$

$\ds \dfrac 1 {\norm y_1 } < 1$

$\ds $

$\leadstoandfrom$

$\ds \norm {y^{-1} }_1 < 1$

$\ds $

$\leadstoandfrom$

$\ds \norm {y^{-1} }_2 < 1$

by assumption

$\ds $

$\leadstoandfrom$

$\ds \dfrac 1 {\norm y_2 } < 1$

$\ds $

$\leadstoandfrom$

$\ds \norm y_2 > 1$

$\blacksquare$