Cauchy Sequence of Subring iff Cauchy Sequence of Normed Division Ring

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 $\struct {R, \norm {\, \cdot \,} }$
 * $\sequence{x_n}$ is a Cauchy sequence in $\struct {R, \norm {\, \cdot \,} }$

Proof
The result follows immediately from: