Cauchy Sequence is Bounded/Normed Division Ring/Proof 2

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

Every Cauchy sequence in $R$ is bounded.


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 $\struct {R, \norm {\,\cdot\,} }$.