Cauchy Sequence is Bounded/Normed Division Ring/Proof 2
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Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Every Cauchy sequence in $R$ is bounded.
Proof
Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n} $ be a Cauchy sequence in $\struct {R, \norm {\,\cdot\,}}$.
By the definition of a Cauchy sequence in a normed division ring then $\sequence {x_n} $ is a Cauchy sequence in $\struct {R, d}$.
By Cauchy sequence is bounded in metric space then $\sequence {x_n} $ is a bounded sequence in $\struct {R, d}$.
By Sequence is Bounded in Norm iff Bounded in Metric then $\sequence {x_n} $ is a bounded sequence in $\struct {R, \norm {\,\cdot\,} }$.
$\blacksquare$
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.2$: Normed Fields, Exercise $11 \ (2)$