Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit

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 $\sequence{x_n}$ be a sequence in $R$.

Let $y \in Q$.

Then:
 * the $\sequence{\map \phi {x_n}}$ converges to $y$ $\sequence{x_n} \in y$

Necessary Condition
Let $\sequence{\map \phi {x_n}}$ converge to $y$.

From Convergent Sequence in Normed Division Ring is Cauchy Sequence:
 * $\sequence{\map \phi {x_n}}$ is a Cauchy Sequence

By definition of a Cauchy Sequence:
 * $\forall \epsilon > 0 : \exists N \in \N : \forall m, n \ge N$:
 * $\norm{\map \phi {x_n} - \map \phi {x_m}}_Q < \epsilon$

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:
 * $\forall r_1, r_2 \in R: \norm{r_1 - r_2}_R = \norm{\map \phi {r_1} - \map \phi {r_2}}_Q$

Hence:
 * $\forall \epsilon > 0 : \exists N \in \N : \forall m, n |ge N$:
 * $\norm{x_n - \map \phi x_m}_R < \epsilon$

It follows that $\sequence{x_n}$ is a Cauchy Sequence by definition.

Let $y'$ be the left coset that contains $\sequence{x'_n}$.

From sufficient condition:
 * $\lim_{n \mathop \to \infty} \norm{\map \phi {x_n} }_Q = y'$

From Convergent Sequence in Metric Space has Unique Limit:
 * $y = y'$

Sufficient Condition
Let $\sequence{x_n} \in y$.

Then $\sequence{x_n}$ is a Cauchy Sequence by definition of $y$.

From Lemma:
 * $\sequence{\map \phi {x_n}}$ is a Cauchy Sequence

From :
 * $Q$ is a complete metric space

From :
 * $\sequence{\map \phi {x_n}}$ converge to some $y' \in Q$

From Necessary Condition:
 * $\sequence{x_n} \in y'$

From Left Cosets Form Partition:
 * $y' = y$

Hence:
 * $\sequence{x_n} \in y$