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.

By Convergent Sequence in Metric Space is Bounded it follows that:
 * $\exists M >0: \forall n, m, \norm {x_n - x_m} \le M$

Then for $n \in \N$, by Axiom (N3) of a norm:
 * $\norm {x_n} = \norm {x_n - x_1 + x_1} \le \norm {x_n - x_1} + \norm {x_1} \le M + \norm {x_1}$

Hence $\sequence {x_n}$ is bounded.