Combination Theorem for Cauchy Sequences

Theorem
Let $\left({R, \left\Vert{\,\cdot\,}\right\Vert}\right)$ be a normed division ring.

Let $\left \langle {x_n} \right \rangle$, $\left \langle {y_n} \right \rangle$ 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