Definition:Equivalent Division Ring Norms/Convergently 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 convergently equivalent metrics

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