Quotient of Cauchy Sequences is Metric Completion/Lemma 1

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 $\mathcal {C} \,\big / \mathcal {N}$ be the quotient ring by the maximal ideal $\mathcal {N}$.

Let $\sim$ be the equivalence relation on $\mathcal C$ defined by:


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

Let $\mathcal C / \sim$ be the set of equivalence classes under $\sim$.

Then:
 * $\quad \mathcal {C} \,\big / \mathcal {N} = \mathcal C / \sim$

Proof
Denote the equivalence class of $\sequence {x_n} \in \mathcal C$ by $\eqclass {x_n} {}$.

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.