Quotient of Cauchy Sequences is Metric Completion

Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $d$ be the metric induced by $\struct {R, \norm {\, \cdot \,} }$.

Let $\mathcal {C}$ be the ring of Cauchy sequences over $R$

Let $\mathcal {N} = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0_R }$

Let $\norm {\, \cdot \,}:\mathcal {C} \,\big / \mathcal {N} \to \R_{\ge 0}$ be the norm on the quotient ring $\mathcal {C} \,\big / \mathcal {N}$ defined by:
 * $\displaystyle \forall \sequence {x_n} + \mathcal {N}: \norm {\sequence {x_n} + \mathcal {N} } = \lim_{n \to \infty} \norm{x_n}$

Let $d'$ be the metric induced by $\struct {\mathcal {C} \,\big / \mathcal {N}, \norm {\, \cdot \,} }$

Then:
 * $\struct {\mathcal {C} \,\big / \mathcal {N}, d'}$ is the metric completion of $\struct {R,d}$

Proof
By Completion Theorem (Metric Space) the metric space constructed as the metric completion of $\struct {R, d}$ is defined by:
 * Let $\mathcal C \left[{R}\right]$ denote the set of all Cauchy sequences in $\struct {R, d}$.


 * Let $\sim$ be the equivalence relation on $\mathcal C \left[{R}\right]$ defined by:


 * $\displaystyle \sequence{x_n} \sim \sequence{y_n} \iff \lim_{n \mathop \to \infty} d \paren{x_n, y_n} = 0$


 * Denote the equivalence class of $\left\langle{x_n}\right\rangle \in \mathcal C \left[{R}\right]$ by $\left[{x_n}\right]$.


 * Denote the set of equivalence classes under $\sim$ by $\mathcal C \left[{R}\right] / \sim$.


 * Define $\tilde d: \mathcal C \left[{R}\right] / \sim \,\to \R_{\ge 0}$ by:


 * $\tilde d \left({\left[{x_n}\right], \left[{y_n}\right]}\right) = \lim_{n \mathop \to \infty} d \left({x_n, y_n}\right)$


 * Then $\struct{\mathcal C \left[{R}\right] / \sim,\tilde d}$ is the metric completion of $\struct {R,d}$.

All that is needed is to show that $\struct {\mathcal {C} \,\big / \mathcal {N}, d'}$ equals $\struct{\mathcal C \left[{R}\right] / \sim,\tilde d}$.

That is, it must be shown that:
 * $\quad \mathcal {C} = \mathcal C \left[{R}\right]$
 * $\quad \mathcal {C} \,\big / \mathcal {N} = \mathcal C \left[{R}\right] / \sim$
 * $\quad d' = \tilde d$

$\quad \mathcal {C} = \mathcal C \left[{R}\right]$
Let $\sequence{x_n}$ be a sequence in $\struct {R, \norm {\, \cdot \,} }$ then:

The result follows.

$\quad \mathcal {C} \,\big / \mathcal {N} = \mathcal C \left[{R}\right] / \sim$
Let $\sequence{x_n}$ and $\sequence{y_n}$ be cauchy sequences in $\mathcal {C} \,\big / \mathcal {N} = \mathcal C \left[{R}\right] / \sim$ then:

The result follows.

$\quad d' = \tilde d$
Let $\sequence{x_n}$ and $\sequence{y_n}$ be cauchy sequences in $\mathcal {C} \,\big / \mathcal {N} = \mathcal C \left[{R}\right] / \sim$ then:

The result follows.