Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 2

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

Let $\CC$ be the ring of Cauchy sequences over $R$.

Let $\NN$ be the set of null sequences.

For all $\sequence {x_n} \in \CC$, let $\eqclass {x_n} {}$ denote the left coset $\sequence {x_n} + \NN$.

Let $\norm {\, \cdot \,}_1: \CC \,\big / \NN \to \R_{\ge 0}$ be defined by:


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

Then:
 * $\norm {\, \cdot \,}_1$ satisfies

That is:
 * $\forall \eqclass {x_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} }_1 = 0 \iff \eqclass {x_n} {} = \eqclass {0_R} {} $

Proof
By Quotient Ring of Cauchy Sequences is Division Ring the zero of $\CC \,\big / \NN$ is $\eqclass {0_R} {}$.

The result follows.