Combination Theorem for Cauchy Sequences

Theorem

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

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

Let $a, b \in R$.

The following results hold:

Constant Rule

The constant sequence $\tuple {a, a, a, \dots}$ is a Cauchy sequence.

Sum Rule

$\sequence {x_n + y_n}$ is a Cauchy sequence.

Difference Rule

$\sequence {x_n - y_n}$ is a Cauchy sequence.

Multiple Rule

$\sequence {a x_n}$ is a Cauchy sequence.

Combined Sum Rule

$\sequence {a x_n + b y_n }$ is a Cauchy sequence.

Product Rule

$\sequence {x_n y_n}$ is a Cauchy sequence.

Inverse Rule

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

Then:

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

and the sequence:

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

Quotient Rule

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

Then:

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

and the sequences:

$\sequence { {x_{K + n} } \paren {y_{K + n} }^{-1} }_{n \mathop \in \N}$ and $\sequence {\paren {y_{K + n} }^{-1} {x_{K + n} } }_{n \mathop \in \N}$ are well-defined and Cauchy sequences.