Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 1

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

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

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

Let $Q = \CC / \NN$ where $\CC  / \NN$ denotes a quotient ring.

Let $\norm {\, \cdot \,}_Q: Q \to \R_{\ge 0}$ be the norm on the quotient ring $Q$ defined by:
 * $\ds \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN }_Q = \lim_{n \mathop \to \infty} \norm{x_n}_R$

Let $\phi: R \to Q$ be the mapping from $R$ to the quotient ring $Q$ defined by:
 * $\forall a \in R: \map \phi a = \tuple {a, a, a, \ldots} + \NN$

where $\sequence {a, a, a, \ldots} + \NN$ is the left coset in $Q$ that contains the constant sequence $\sequence {a, a, a, \ldots} $.

Let $\sequence{y_n}$ be a sequence in $R$.

Then:
 * $\sequence{y_n}$ is a Cauchy sequence $\sequence{\map \phi {y_n}}$ is a Cauchy sequence

Proof
From Embedding Division Ring into Quotient Ring of Cauchy Sequences:
 * the mapping $\phi: R \to Q$ is a distance-preserving monomorphism

By definition of distance-preserving mapping:
 * $\forall r_1, r_2 \in R: \norm{r_1 - r_2}_R = \norm{\map \phi {r_1} - \map \phi {r_2}}_Q$

By definition of a Cauchy Sequence:
 * $\sequence{y_n}$ is a Cauchy sequence


 * $\forall \epsilon > 0 : \exists N \in \N : \forall m, n \ge N$: $\norm{y_n - y_m}_Q < \epsilon$
 * $\forall \epsilon > 0 : \exists N \in \N : \forall m, n \ge N$: $\norm{y_n - y_m}_Q < \epsilon$


 * $\forall \epsilon > 0 : \exists N \in \N : \forall m, n \ge N$: $\norm{\map \phi {y_n} - \map \phi {y_m}}_Q < \epsilon$
 * $\forall \epsilon > 0 : \exists N \in \N : \forall m, n \ge N$: $\norm{\map \phi {y_n} - \map \phi {y_m}}_Q < \epsilon$


 * $\sequence{\map \phi {y_n}}$ is a Cauchy sequence:
 * $\sequence{\map \phi {y_n}}$ is a Cauchy sequence: