Cauchy Sequence of Subring iff Cauchy Sequence of Normed Division Ring
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Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\struct {S, \norm {\, \cdot \,}_S }$ be a normed division subring of $\struct {R, \norm {\, \cdot \,} }$.
Let $\sequence{x_n}$ be a sequence in $S$.
Then:
- $\sequence{x_n}$ is a Cauchy sequence in $\struct {S, \norm {\, \cdot \,}_S }$
- $\sequence{x_n}$ is a Cauchy sequence in $\struct {R, \norm {\, \cdot \,} }$
Proof
The result follows immediately from:
- Definition of Cauchy Sequence (Normed Division Ring)
- Definition of Normed Division Subring
$\blacksquare$