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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 Preview

    Please see the attachment.

    Problem #1
    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 ...

    Solution Summary

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