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 $\CC$ 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:
- $\ds \forall \eqclass {x_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} }_1 = \lim_{n \mathop \to \infty} \norm {x_n}$
Then:
- $\norm {\, \cdot \,}_1$ satisfies the Norm Axiom $\text N 3$: Triangle Inequality.
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 $\eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN$
\(\ds \norm {\eqclass {x_n} {} + \eqclass {y_n} {} } _1\) | \(=\) | \(\ds \norm {\eqclass {x_n + y_n} {} }_1\) | Addition on quotient ring | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \norm {x_n + y_n}\) | Definition of $\norm {\,\cdot\,}_1$ |
By Norm Axiom $\text N 3$: Triangle Inequality:
- $\forall n: \norm {x_n + y_n} \le \norm {x_n} + \norm {y_n}$
So:
\(\ds \lim_{n \mathop \to \infty} \norm {x_n + y_n}\) | \(\le\) | \(\ds \lim_{n \mathop \to \infty} \norm {x_n} + \norm {y_n}\) | Inequality Rule for Real Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \norm {x_n} + \lim_{n \mathop \to \infty} \norm {y_n}\) | Sum Rule for Real Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm {\eqclass {x_n} {} }_1 + \norm {\eqclass {y_n} {} } _1\) | Definition of $\norm {\,\cdot\,}_1$ |
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions