Combination Theorem for Sequences/Normed Division Ring/Quotient Rule

Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero: $0$.

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

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


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

Suppose $m \ne 0$.

Then:
 * $\exists k \in \N : \forall n \in \N: y_{k + n} \ne 0$

and the sequences:
 * $\sequence {x_{k + n} \ {y_{k + n} }^{-1} }$ and $\sequence { {y_{k + n} }^{-1} \ x_{k + n} }$ are well-defined and convergent with:


 * $\displaystyle \lim_{n \mathop \to \infty} x_{k + n} \ {y_{k + n} }^{-1} = l m^{-1}$
 * $\displaystyle \lim_{n \mathop \to \infty} {y_{k + n} }^{-1} \ x_{k + n} = m^{-1} l$

Proof
By the Inverse Rule for Normed Division Ring:
 * $\exists k \in \N : \forall n \in \N : y_{k + n} \ne 0$

and the sequence:
 * $\sequence { {y_{k + n} }^{-1} }$

is well-defined and convergent with:
 * $\displaystyle \lim_{n \mathop \to \infty} {y_{k + n} }^{-1} = m^{-1}$

By Limit of Subsequence equals Limit of Sequence, $\sequence {x_{k + n} }$ is convergent with:
 * $\displaystyle \lim_{n \mathop \to \infty} x_{k + n} = l$

By Product Rule for Normed Division Ring Sequences:
 * $\displaystyle \lim_{n \mathop \to \infty} x_{k + n} \ {y_{k + n} }^{-1} = l m^{-1}$
 * $\displaystyle \lim_{n \mathop \to \infty} {y_{k + n} }^{-1} \ x_{k + n} = m^{-1} l$