Equivalence of Definitions of Equivalent Division Ring Norms/Cauchy Sequence Equivalent implies Open Unit Ball Equivalent

Theorem
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 $\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ satisfy:
 * for all sequences $\sequence {x_n}$ in $R:\sequence {x_n}$ is a Cauchy sequence in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ is a Cauchy sequence in $\norm{\,\cdot\,}_2$

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

Proof
The contrapositive is proved.

Let there exist $x \in R$ such that $\norm{x}_1 \lt 1$ and $\norm{x}_2 \ge 1$.

Let

Let $0_R$ be the zero of $R$ and $1_R$ be the unit of $R$.

By norm of unity and the assumption $\norm{x}_1 \lt 1$ then $x \neq 1_R$.

Then $1_R - x \neq 0_R$.

By norm axiom (N1) (Positive Definiteness) then $\norm {1_R - x}_2 \gt 0$.

Let $\epsilon = $\norm {1_R - x}_2$.