Quotient Ring of Cauchy Sequences is Normed Division Ring

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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:

$\struct {\CC \,\big / \NN, \norm {\, \cdot \,}_1 }$ is a normed division ring.


Corollary

Let $\struct {R, \norm {\, \cdot \,} }$ be a valued field.


Then $\struct {\CC \,\big / \NN, \norm {\, \cdot \,}_1 }$ is a valued field.


Proof

By Quotient Ring of Cauchy Sequences is Division Ring then $\CC \,\big / \NN$ is a division ring.

It remains to be proved that:

$\norm {\, \cdot \,}_1$ is well-defined
$\norm {\, \cdot \,}_1$ satisfies the norm axioms.


Lemma 1

$\norm {\, \cdot \,}_1$ is well-defined.

That is,

$(1): \quad \ds \forall \eqclass {x_n}{}: \lim_{n \mathop \to \infty} \norm{x_n}$ exists
$(2): \quad \ds \forall \eqclass {x_n}{}, \eqclass {y_n}{} \in \CC \,\big / \NN: \eqclass {x_n}{} = \eqclass {y_n}{} \implies \lim_{n \mathop \to \infty} \norm{x_n} = \lim_{n \mathop \to \infty} \norm{y_n}$

$\Box$


Lemma 2

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

$\Box$


Lemma 3

$\norm {\, \cdot \,}_1$ satisfies the Norm Axiom $\text N 2$: Multiplicativity.

That is:

$\forall \eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} \eqclass {y_n} {} }_1 = \norm {\eqclass {x_n} {} }_1 \times \norm {\eqclass {y_n} {} }_1$

$\Box$


Lemma 4

$\norm {\, \cdot \,}_1$ satisfies the Norm Axiom $\text N 3$: Triangle Inequality.

That is:

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

$\Box$


The result follows.

$\blacksquare$


Sources

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