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:

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

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