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:


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

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.

It remains to be proved that:
 * $\norm {\, \cdot \,}_1$ is well-defined
 * $\norm {\, \cdot \,}_1$ satisfies the norm axioms.

Lemma 4
The result follows.