Equivalence of Definitions of Equivalent Division Ring Norms/Null Sequence Equivalent implies Open Unit Ball 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:
 * $\forall x \in R: \norm{x}_1 \lt 1 \iff \norm{x}_2 \lt 1$

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