Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 2

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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 $\tuple {a, a, a, \ldots} + \NN$ 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 $y$ be the left coset in $Q$ that contains $\sequence{x_n}$.


For each $n \in \N$:

$\norm{\map \phi {x_n} - y}_Q = \ds \lim_{m \mathop \to \infty} \norm{x_n - x_m}_R$


Proof

Let $n \in \N$.

We have:

$\tuple {x_n, x_n, x_n, \ldots} \in \map \phi {x_n}$
$\sequence{x_m} \in y$

From Element of Group is in Unique Coset of Subgroup:

$\tuple {x_n, x_n, x_n, \ldots} + \NN = \map \phi {x_n}$
$\sequence{x_m} + \NN = y$


Then:

\(\ds \sequence{x_n - x_m}_{m \in \N} + \NN\) \(=\) \(\ds \paren{\tuple {x_n, x_n, x_n, \ldots} - \sequence{x_m}_{m \in \N} } + \NN\) Difference Rule for Cauchy Sequences in Normed Division Ring
\(\ds \) \(=\) \(\ds \paren{\tuple {x_n, x_n, x_n, \ldots} + \NN} - \paren{\sequence{x_m}_{m \in \N} + \NN}\) Definition of Quotient Ring
\(\ds \) \(=\) \(\ds \map \phi {x_n} - y\)

From Element of Group is in its own Coset:

$\sequence{x_n - x_m}_{m \in \N} \in \map \phi {x_n} - y$


By definition of induced norm:

$\norm{\map \phi {x_n} - y}_Q = \ds \lim_{m \mathop \to \infty} \norm{x_n - x_m}_R$

$\blacksquare$

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