Normed Division Ring Completions are Isometric and Isomorphic/Lemma 2

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Theorem

Let $\struct {S_1, \norm {\, \cdot \,}_1 }$ and $\struct {S_2, \norm {\, \cdot \,}_2 }$ be complete normed division rings.

Let $R_1$ be a dense subring of $S_1$.

Let $R_2$ be a dense subring of $S_2$.

Let $\psi': R_1 \to R_2$ be an isometric ring isomorphism.

Let $\psi: S_1 \to S_2$ be defined by:

$\forall x \in S_1: \map \psi x = \ds \lim_{n \mathop \to \infty} \map {\psi'} {x_n}$

where $\ds x = \lim_{n \mathop \to \infty} x_n$ for some sequence $\sequence {x_n} \subseteq R_1$

Then $\psi$ is a well-defined mapping.


Proof

Let $x \in S_1$.

By the definition of dense subset:

$\map \cl {R_1} = S_1$

By Closure of Subset of Metric Space by Convergent Sequence, there exists a sequence $\sequence {x_n} \subseteq R_1 $ that converges to $x$, that is:

$\ds \lim_{n \mathop \to \infty} x_n = x$

By Isometric Image of Cauchy Sequence is Cauchy Sequence, $\sequence {\map {\psi'} {x_n} }$ is a Cauchy sequence in $R_2 \subseteq S_2$.

Since $S_2$ is complete then the sequence $\sequence {\map {\psi'} {x_n} }$ has a limit, say $y$.


Let $\sequence {x_n}$ and $\sequence {x'_n}$ be sequences in $\map {\phi_1} R$ such that:

$\ds \lim_{n \mathop \to \infty} x_n = \lim_{n \mathop \to \infty} x'_n = x$


Then:

\(\ds \lim_{n \mathop \to \infty} x_n - x'_n\) \(=\) \(\ds \paren {\lim_{n \mathop \to \infty} x_n} - \paren {\lim_{n \mathop \to \infty} x'_n}\) Difference Rule for Sequences in Normed Division Ring
\(\ds \) \(=\) \(\ds x - x\)
\(\ds \) \(=\) \(\ds 0_{S_1}\)
\(\ds \) \(\) \(\ds \)
\(\ds \leadsto \ \ \) \(\ds 0\) \(=\) \(\ds \lim_{n \mathop \to \infty} \norm {x_n - x'_n}_1\) Definition of Convergent Sequence in Normed Division Ring
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \norm {\map {\psi'} {x_n} - \map {\psi'} {x'_n} }_2\) Definition of Isometric Metric Spaces
\(\ds \) \(\) \(\ds \)
\(\ds \leadsto \ \ \) \(\ds 0_{S_2}\) \(=\) \(\ds \lim_{n \mathop \to \infty} \map {\psi'} {x_n} - \map {\psi'} {x'_n}\) Definition of Convergent Sequence in Normed Division Ring
\(\ds \) \(=\) \(\ds \paren {\lim_{n \mathop \to \infty} \map {\psi'} {x_n} } - \paren {\lim_{n \mathop \to \infty} \map {\psi'} {x'_n} }\) Difference Rule for Sequences in Normed Division Ring
\(\ds \) \(\) \(\ds \)
\(\ds \leadsto \ \ \) \(\ds \lim_{n \mathop \to \infty} \map {\psi'} {x_n}\) \(=\) \(\ds \lim_{n \mathop \to \infty} \map {\psi'} {x'_n}\)

The result follows.

$\blacksquare$

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