Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Necessary Condition

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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 $y \in Q$.

If $\sequence{\map \phi {x_n}}$ converges to $y$ then $\sequence{x_n} \in y$

Proof

Lemma 1

$\sequence{x_n}$ is a Cauchy sequence if and only if $\sequence{\map \phi {x_n}}$ is a Cauchy sequence

$\Box$

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

From Lemma 1:

$\sequence{x_n}$ is a Cauchy Sequence

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

From sufficient condition:

$\ds \lim_{n \mathop \to \infty} \norm{\map \phi {x_n} }_Q = y'$

From Convergent Sequence in Metric Space has Unique Limit:

$y = y'$

$\blacksquare$