\begin{definition}[Definition:Equivalent Division Ring Norms/Cauchy Sequence Equivalent]
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.
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$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent''' {{iff}} $d_1$ and $d_2$ are Cauchy equivalent metrics.
That is, $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent''' {{iff}} 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 $\norm{\,\cdot\,}_2$
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\end{definition}