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Cauchy and sequences

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1. Let ( ) be a bounded sequence. Show that there exists a subsequence of ( ) converging to

2. Show that is not a Cauchy sequence Conclude that diverges.

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Solution Summary

This is a proof regarding subsequences of a bounded sequence and another proof regarding a Cauchy sequence.

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Problem #1
Proof:
Let , then is a monotone increasing sequence. Since is a bounded sequence, then is bounded. Thus has a limit. So we have

We also have for any .
We have two cases:
Case 1: We can find some , such that for all ...

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