Combination Theorem for Cauchy Sequences/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 Cauchy sequences in $R$.

Suppose $\sequence {y_n}$ does not converge to $0$.

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 Cauchy sequences.


Proof

By the Inverse Rule for Normed Division Ring:

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

and the sequence

$\sequence { \paren {x_{K+n}}^{-1} }_{n \in \N}$ is well-defined and a Cauchy sequence.

By Subsequence of a Cauchy Sequence is a Cauchy Sequence then $\sequence { x_{K+n} }_{n \in \N}$ is a Cauchy sequence.

By Product Rule for Normed Division Ring Sequences:

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 Cauchy sequences.

$\blacksquare$


Sources