Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 4

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 the.

That is:
 * $\forall \eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} + \eqclass {y_n} {} }_1 \le \norm {\eqclass {x_n} {} }_1 + \norm {\eqclass {y_n} {} }_1$

Proof
Let $\eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN$

By :
 * $\forall n: \norm {x_n + y_n} \le \norm {x_n} + \norm {y_n}$

So: