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 $R$.

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

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

Then:
 * $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ are equivalent $d_1$ and $d_2$ are convergently equivalent metrics