# Convergent Sequence is Cauchy Sequence/Normed Division Ring/Proof 2

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## Contents

## Theorem

Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring.

Every convergent sequence in $R$ is a Cauchy sequence.

## Proof

Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$.

Let $\sequence {x_n}$ be a sequence in $R$ that converges to the limit $l$ in $\struct {R, \norm {\,\cdot\,}}$.

Thus, by definition, $\sequence {x_n} $ converges to the limit $l$ in $\struct {R, d}$.

By Convergent Sequence is Cauchy Sequence in metric space, $\sequence {x_n} $ is a Cauchy sequence in $\struct {R, d}$.

Thus, by definition, $\sequence {x_n} $ is a Cauchy sequence in $\struct {R, \norm {\,\cdot\,}}$.

$\blacksquare$

## Also see

## Sources

- 2007: Svetlana Katok:
*p-adic Analysis Compared with Real*: $\S 1.2$: Normed Fields, Exercise $11 \ (1)$