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

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 Norm Axiom $\text N 1$: Positive Definiteness

That is:

$\forall \eqclass {x_n} {} \in \CC \,\big / \NN: \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 $\CC \,\big / \NN$ is $\eqclass {0_R} {}$.

\(\ds \norm {\eqclass {0_R} {} }_1 = 0\)

\(\leadstoandfrom\)

\(\ds \lim_{n \mathop \to \infty} \norm {x_n} = 0\)

Definition of $\norm {\,\cdot\,}_1$

\(\ds \)

\(\leadstoandfrom\)

\(\ds \sequence {x_n} \in \NN\)

Definition of $\NN$

\(\ds \)

\(\leadstoandfrom\)

\(\ds \eqclass {x_n} {} = \eqclass {0_R} {}\)

Left Cosets are Equal iff Product with Inverse in Subgroup

The result follows.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions