Convergent Sequence in Normed Division Ring is Bounded

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

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

Let $\sequence {x_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limit:


 * $\displaystyle \lim_{n \mathop \to \infty} x_n = l$

Then $\sequence {x_n}$ is bounded.

Proof
From Metric Induced by Norm on a Normed Division Ring is Metric, the set $R$ under the norm metric is a metric space.

Hence Convergent Sequence in Metric Space is Bounded can be applied directly.