# 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 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 |

$\blacksquare$

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*: $\S 3.1$ Absolute Values on $\Q$, Lemma 3.1.2 and Problem 66