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 $\mathcal N$ be the set of null sequences.

Let $\mathcal C \,\big / \mathcal N$ be the quotient ring of Cauchy sequences of $\mathcal C$ by the maximal ideal $\mathcal N$.

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


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

Let $\tilde {\mathcal C} = \mathcal C / \sim$ denote the set of equivalence classes under $\sim$.

For $\sequence {x_n} \in \mathcal C$, let $\eqclass {x_n} {}$ denote the equivalence class containing $\sequence {x_n}$.

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

Proof
Let $\sequence {x_n}$ and $\sequence {y_n}$ be Cauchy sequences in $\mathcal C$.

Then:

Hence:
 * $\sequence {x_n}$ and $\sequence {y_n}$ belong to the same equivalence class in $\mathcal C \,\big / \mathcal N$ $\sequence {x_n}$ and $\sequence {y_n}$ belong to the same equivalence class in $\tilde {\mathcal C}$.

The result follows.