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Advanced Calculus Analysis : Cauchy and Convergent Sequences

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We proved the following theorem in the class: "If a>0 and if a sequence... is covergent, then the sequence... is convergent." In proving this theorem, we proved that... is Cauchy instead of proving it converges directly. Why did we have to do that?

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Cauchy and Convergent Sequences are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.

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Please see the attachment.

Theorem: Suppose and the sequence converges. Show that the sequence converges.

Proof:
First, I claim that if , then . This is true because is a ...

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