Combination Theorem for Sequences/Normed Division Ring/Quotient Rule

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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 then $\sequence { x_{k+n} }$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} x_{k+n} = l$.

By Product Rule for Normed Division Ring Sequences:

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

$\blacksquare$


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