Definition:Equivalent Division Ring Norms/Cauchy Sequence Equivalent

Definition
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.

 $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ are equivalent $d_1$ and $d_2$ are Cauchy equivalent metrics.

That is, $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ are equivalent for all sequences $\sequence {x_n}$ in $R$:
 * $\sequence {x_n}$ is a Cauchy sequence in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ is a Cauchy sequence in $\norm{\,\cdot\,}_2$