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 $\CC$ be the ring of Cauchy sequences over $R$

Let $\NN$ be the set of null sequences.

Let $\CC \,\big / \NN$ be the quotient ring of Cauchy sequences of $\CC$ by the maximal ideal $\NN$.

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


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

Let $\tilde \CC = \CC / \sim$ denote the set of equivalence classes under $\sim$.

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

Then:
 * $\quad \CC \,\big / \NN = \tilde \CC$

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

Then:

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

The result follows.