Combination Theorem for Cauchy Sequences

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

Let $\sequence {x_n}$, $\sequence {y_n} $ be Cauchy sequences in $R$.

Then the following results hold:

Quotient Rule
If $R$ is also a commutative ring, that is, $\left({R, \left\Vert{\,\cdot\,}\right\Vert}\right)$ is a valued field, then