Combination Theorem for Cauchy Sequences
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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.
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions