Sequence is Bounded in Norm iff Bounded in Metric

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

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

Let $\sequence {x_n}$ be a sequence in $R$.

Then:
 * $\sequence {x_n} $ is a bounded sequence in the normed division ring $\struct {R, \norm {\,\cdot\,}}$ $\sequence {x_n} $ is a bounded sequence in the metric space $\struct {R, d}$

Also See

 * Bounded sequence in a normed division ring
 * Metric induced by a norm on a division ring
 * Bounded sequence in a metric space