Calculus Small Changes, Big Ideas

Section 22.4 Cauchy sequences

• A sequence is a Cauchy sequence if for all \(\eps>0\) there exists some \(N\in\bb N\) such that \(|a_{m}-a_{n}|<\eps\) for all \(m,n>N\text{.}\)

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• If a sequence is Cauchy, then it is bounded.

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• If a sequence converges, then it is Cauchy.

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• If a sequence in a complete space is Cauchy, then it converges. This property is called Cauchy completeness.

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• \(\eps/2\) trick

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• “Ratio test”: If \((a_{n})\) is positive and \(\limit{n\to\infty}\dfrac{a_{n+1}}{a_{n}}\text{,}\) then \(\limit{n\to\infty}a_{n}=0\text{.}\)

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• Monotone convergence theorem: A monotone sequence of real numbers converges if and only if it is bounded. Furthermore, the limit is the supremum or infimum. (Could apply to definition of \(\ee\text{.}\))

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