Equivalence of Definitions of Equivalent Division Ring Norms/Norm is Power of Other Norm implies Topologically Equivalent

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 $d_1$ and $d_2$ be the metrics induced by the norms $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ respectively.

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy:
 * $\exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$

Then $d_1$ and $d_2$ are topologically equivalent metrics.

Proof
Let $x \in R$ and $\epsilon \in \R_{\gt 0}$

Then for $y \in R$:

Hence:
 * $\map {B^1_\epsilon} x = \map {B^2_{\epsilon^{1 / \alpha} } } x$

where:
 * $\map {B^1_\epsilon} x$ is the open ball in $d_1$ centered on $x$ with radius $\epsilon$
 * $\map {B^2_{\epsilon^{1 / \alpha} } } x$ is the open ball in $d_2$ centered on $x$ with radius $\epsilon^{1 / \alpha}$

Since $x$ and $\epsilon$ were arbitrary then:
 * every $d_1$-open ball is a $d_2$-open ball.

Similarly, for $y \in R$:

So:
 * every $d_2$-open ball is a $d_1$-open ball.

By the definition of an open set of a metric space it follows that $d_1$ and $d_2$ are topologically equivalent metrics,