Equivalence of Definitions of Equivalent Division Ring Norms/Convergently Equivalent implies Null Sequence Equivalent

Theorem
Let $\struct{R, \norm{\,\cdot\,}_1}$ and $\struct{R, \norm{\,\cdot\,}_2}$ be normed division rings on the same underlying division ring $R$. Let $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ satisfy:
 * 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 $\norm{\,\cdot\,}_2$

Then:
 * $\forall x \in R: \norm{x}_1 \lt 1 \iff \norm{x}_2 \lt 1$