Cauchy Sequence is Bounded/Normed Division Ring/Proof 3

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

Every Cauchy sequence in $R$ is bounded.

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

By Norm Sequence of Cauchy Sequence has Limit 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:
 * $\exists M \in \R_{\gt 0}: \forall n, \norm {x_n} = \left \vert { \norm {x_n} } \right \vert \le M$

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