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Math Help - prove/counterexample for sequences

  1. #1
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    prove/counterexample for sequences

    Prove or give a counter example:

    A) Every bounded sequence has a Cauchy subsequence
    B) Every monotone sequence has a bounded subsequence
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  2. #2
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    Quote Originally Posted by learn18 View Post
    Prove or give a counter example:

    A) Every bounded sequence has a Cauchy subsequence
    Every Bounded Sequence has a convergent subsequence.
    (See Bolzano-Weierstrass Theorem).
    But a convergent sequence is a Cauchy sequence.

    B) Every monotone sequence has a bounded subsequence
    No consider,
    s_n=n
    We see that every subsequence must diverge to +oo.

    Hence all sequences diverge to +oo.
    (Excercise).

    But then no subsequence is bounded because it diverges to +oo.
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