Combination Theorem for Sequences/Normed Division Ring

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

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

Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limits:


 * $\displaystyle \lim_{n \mathop \to \infty} x_n = l$
 * $\displaystyle \lim_{n \mathop \to \infty} y_n = m$

Let $\lambda, \mu \in R$.

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