Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition

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 $R$.

Let $\sequence {x_n} $ be a bounded sequence in the normed division ring $\struct {R, \norm {\,\cdot\,}}$

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

Proof
Let $\sequence {x_n} $ be a bounded sequence in $\struct {R, \norm {\,\cdot\,} }$.

Then:
 * $\exists K \in \R_{\gt 0} : \forall n : \norm {x_n} \le K$

Then $\forall n, m \in \N$:

Hence the sequence $\sequence {x_n} $ is bounded by $2 K$ in the metric space $\struct {R, d}$.

Also see

 * Definition:Bounded Sequence in Normed Division Ring
 * Definition:Metric Induced by Norm on Division Ring
 * Definition:Bounded Sequence in Metric Space