Equivalence of Definitions of Equivalent Division Ring Norms/Convergently Equivalent implies Null Sequence 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 Convergent sequence in $\norm{\,\cdot\,}_1$ with limit $l \iff \sequence {x_n}$ is a Convergent sequence in $\norm{\,\cdot\,}_2$ with limit $l$

Then 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 $\norm{\,\cdot\,}_2$

Proof
Let $0_R$ be the zero of $R$, then:
 * $\sequence {x_n}$ converges to $0_R$ in $\norm{\,\cdot\,}_1 \iff \sequence {x_n}$ is a converges to $0_R$ in $\norm{\,\cdot\,}_2$

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