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

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

Also see

 * Cauchy's Convergence Criterion on Real Numbers
 * Cauchy's Convergence Criterion on Complex Numbers