Definition:Equivalent Division Ring Norms/Null 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 $R$.

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