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:
 * $\exists K \in \N : \forall n > K : y_n \ne 0$.

and the sequences
 * $\sequence { {x_{K+n}} \paren {y_{K+n}}^{-1} }_{n \in \N}$ and $\sequence { \paren {y_{K+n}}^{-1} {x_{K+n}} }_{n \in \N}$ are well-defined and convergent with:


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

Proof
By the Inverse Rule for Normed Division Ring:
 * $\exists K \in \N : \forall n > K : y_n \ne 0$.

and the sequence
 * $\sequence { \paren {y_{K+n}}^{-1} }_{n \in \N}$ is well-defined and convergent with $\displaystyle \lim_{n \mathop \to \infty} \paren {y_{K+n} }^{-1} = m^{-1}$.

By Limit of Subsequence equals Limit of Sequence then $\sequence { x_{K+n} }_{n \in \N}$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} x_{K+n} = l$.

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