Equivalence of Definitions of Equivalent Division Ring Norms/Null 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 null sequence in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_2$

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

Proof
Let $x \in R$.

Let $\sequence {x_n}$ be the Definition:Sequence defined by: $\forall n: x_n = x^n$.

By Sequence of Powers of Number less than One in Normed Division Ring then:
 * $\norm{x}_1 \lt 1 \iff \sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_1$.

By assumption:
 * $\sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_2$

By Sequence of Powers of Number less than One in Normed Division Ring then:
 * $\sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_1 \iff \norm{x}_1 \lt 1$

The result follows.