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

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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:

$\norm {\, \cdot \,}_1$ satisfies the norm axiom (N3).

That is:

$\forall \eqclass {x_n}{}, \eqclass {y_n}{} \in \mathcal {C} \,\big / \mathcal {N}: \norm {\eqclass {x_n}{} + \eqclass {y_n}{} }_1 \le \norm {\eqclass {x_n}{} }_1 + \norm {\eqclass {y_n}{} }_1$


Proof

Let $\eqclass {x_n}{}, \eqclass {y_n}{} \in \mathcal {C} \,\big / \mathcal {N}$

\(\displaystyle \norm { \eqclass {x_n}{} + \eqclass {y_n}{} } _1\) \(=\) \(\displaystyle \norm { \eqclass {x_n + y_n}{} }_1\) Addition on quotient ring
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \to \infty} \norm { x_n + y_n }\) Definition of $\norm {\,\cdot\,}_1$

By norm axiom (N3) (Triangle Inequality) then:

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

So:

\(\displaystyle \lim_{n \to \infty} \norm { x_n + y_n }\) \(\le\) \(\displaystyle \lim_{n \to \infty} \norm { x_n } + \norm {y_n }\) Inequality Rule for Real Sequences
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \to \infty} \norm { x_n } + \lim_{n \to \infty} \norm {y_n }\) Sum Rule for Real Sequences
\(\displaystyle \) \(=\) \(\displaystyle \norm { \eqclass {x_n}{} }_1 + \norm { \eqclass {y_n}{} } _1\) Definition of $\norm {\,\cdot\,}_1$

$\blacksquare$


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