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

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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 sequence defined by: $\forall n: x_n = x^n$.


\(\displaystyle \norm{x}_1 \lt 1 \quad\) \(\iff\) \(\displaystyle \) $\sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_1$ Sequence of Powers of Number less than One in Normed Division Ring
\(\displaystyle \) \(\iff\) \(\displaystyle \) $\sequence {x_n}$ is a null sequence in $\norm{\,\cdot\,}_2$ Assumption
\(\displaystyle \) \(\iff\) \(\displaystyle \) $\norm{x}_2 \lt 1$ Sequence of Powers of Number less than One in Normed Division Ring


$\blacksquare$

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