Convergent Sequence in Normed Division Ring is Bounded/Proof 3

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
Let $\sequence {x_n}$ be convergent to the limit $l$ in $\struct {R, \norm {\,\cdot\,}}$.

By modulus of limit in normed division ring then $\sequence {\norm {x_n} }$ is a convergent sequence in $\R$.

By Convergent Real Sequence is Bounded then $\sequence {\norm {x_n} }$ is bounded.

That is:
 * $\exists M \in \R_{\gt 0}: \forall n, \norm {x_n} = \size { \norm {x_n} } \le M$

By the definition of a bounded sequence in a normed division ring then $\sequence {x_n}$ is bounded.