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Math Help - Cauchy subsequence and a limit

  1. #1
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    Cauchy subsequence and a limit

    1)
    prove or give a counterexample:
    A) every bounded sequence has a Cauchy subsequence
    B) every monotone sequence has a bounded subsequence

    2)
    Suppose that x>1. Prove that Lim x^(1/n) = 1
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  2. #2
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    Quote Originally Posted by slowcurv99 View Post
    1)
    prove or give a counterexample:
    A) every bounded sequence has a Cauchy subsequence
    B) every monotone sequence has a bounded subsequence

    A)Every bounded sequence has a convergent subsequence. That sequence is a Cauchy sequence.
    B)No. s_n=n.
    2)
    Suppose that x>1. Prove that Lim x^(1/n) = 1
    1<= x <= n for sufficiently large n.
    Then,
    1<=x^{1/n} <= n^{1/n}
    Squeeze theorem.

    Use famous fact that,
    n^{1/n} --> 1
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