Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring/Corollary

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Theorem

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

Let $\struct {S, \norm {\, \cdot \,}}$ be a dense normed division subring of $\struct {R, \norm {\, \cdot \,}}$.


Then for all $x \in R$, there exists a sequence $\sequence{x_n}$ in $S$:

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


Proof

From Leigh.Samphier/Sandbox/Complete Normed Division Ring is Completion of Dense Subring:

$\struct {R, \norm {\, \cdot \,} }$ is a completion of $\struct {S, \norm {\, \cdot \,}}$

where the inclusion mapping $i : S \to R$ is the required distance-preserving ring monomorphism.


From Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring:

for all $x \in R$, there exists a sequence $\sequence{x_n}$ in $S$:
$\quad x = \displaystyle \lim_{n \mathop \to \infty} \map i {x_n}$

That is:

for all $x \in R$, there exists a sequence $\sequence{x_n}$ in $S$:
$\quad x = \displaystyle \lim_{n \mathop \to \infty} x_n$

$\blacksquare$