Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion/Corollary

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 $S$:

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

and

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

Proof

$\blacksquare$