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

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

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

Let $\mathcal {N} = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0 }$

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:
 * $\norm {\, \cdot \,}_1$ satisfies the norm axiom (N1).

That is:
 * $\forall \eqclass {x_n}{} \in \mathcal {C} \,\big / \mathcal {N}: \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 $\mathcal {C} \,\big / \mathcal {N}$ is $\eqclass {0_R}{}$.

The result follows.