Quotient of Cauchy Sequences is Metric Completion/Lemma 2

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

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

Let $\struct {\mathcal {C} \,\big / \mathcal {N}, \norm {\, \cdot \,} }$ be the normed quotient ring of Cauchy sequences over $R$ where:
 * $\mathcal {N}$ is the maximal ideal: $\quad \mathcal {N} = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0_R }$

and
 * $\norm {\, \cdot \,}$ is the norm on the quotient ring $\mathcal {C} \,\big / \mathcal {N}$ defined by: $\quad \displaystyle \forall \sequence {x_n} + \mathcal {N}: \norm {\sequence {x_n} + \mathcal {N} } = \lim_{n \to \infty} \norm{x_n}$

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

Let $\struct{\mathcal C / \sim,\tilde d}$ is the metric completion of $\struct {R,d}$ where:


 * $\sim$ is the equivalence relation on $\mathcal C$ defined by: $\quad \displaystyle \sequence{x_n} \sim \sequence{y_n} \iff \lim_{n \mathop \to \infty} d \paren{x_n, y_n} = 0$

and


 * $\tilde d$ is the metric on $\mathcal C / \sim$ defined by: $\quad \map {\tilde d} {\eqclass {x_n}{}, \eqclass {y_n}{}} = \lim_{n \mathop \to \infty} \map d {x_n, y_n}$

Then:
 * $\quad d' = \tilde d$

Proof
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.