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

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

By the definition of a convergent sequence in a normed division ring then $\sequence {x_n} $ converges to the limit $l$ in $\struct {R, d}$.

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

By the definition of a Cauchy sequence in a normed division ring then $\sequence {x_n} $ is a Cauchy sequence in $\struct {R, \norm {\,\cdot\,}}$.

Also see

 * Real Convergent Sequence is Cauchy Sequence
 * Complex Sequence is Cauchy iff Convergent