Explore BrainMass

Explore BrainMass

    Cauchy and sequences

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    See attachment

    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.

    © BrainMass Inc. brainmass.com November 30, 2021, 2:32 am ad1c9bdddf
    https://brainmass.com/math/algebra/cauchy-bounded-sequences-184438

    Attachments

    Solution Preview

    Please see the attachment.

    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 ...

    Solution Summary

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

    $2.49

    ADVERTISEMENT