Quotient Ring of Cauchy Sequences is Normed Division Ring

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

Let $\mathcal {C}$ be the ring of Cauchy sequences over $R$

Let $\mathcal {N}$ be the set of null sequences.

For all $\sequence {x_n} \in \mathcal {C}$, let $\eqclass {x_n}{}$ denote the left coset $\sequence {x_n} + \mathcal {N}$

Let $\norm {\, \cdot \,}_1:\mathcal {C} \,\big / \mathcal {N} \to \R_{\ge 0}$ be defined by:


 * $\displaystyle \forall \eqclass {x_n}{} \in \mathcal {C} \,\big / \mathcal {N}: \norm {\eqclass {x_n}{} }_1 = \lim_{n \to \infty} \norm{x_n}$

Then:
 * $\struct {\mathcal {C} \,\big / \mathcal {N}, \norm {\, \cdot \,}_1 }$ is a normed division ring.

Corollary
Let $\struct {R, \norm {\, \cdot \,} }$ be a valued field.

Proof
By Quotient Ring of Cauchy Sequences is Division Ring then $\mathcal {C} \,\big / \mathcal {N}$ is a division ring and it only remains to be proved that $\norm{\,\cdot\,}_1$ is well-defined and satisfies the norm axioms.