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Math Help - Convergent subsequences

  1. #1
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    Convergent subsequences

    Which of these sequences b_n always has a convergent subsequence, regardless of what a_n is? Indicate reasoning.
    For those subsequences b_n which do not always have a convergent subsequence, give a example; that is produce a sequence a_n such that the corresponding b_n has no convergent subsequence.

    (a) b_n = cos^2(a_n) (b) bn = a_n/(1 + a_n) (c) b_n = 1/(1 + |a_n|)
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  2. #2
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    Quote Originally Posted by cgiulz View Post
    Which of these sequences b_n always has a convergent subsequence, regardless of what a_n is? Indicate reasoning.
    For those subsequences b_n which do not always have a convergent subsequence, give a example; that is produce a sequence a_n such that the corresponding b_n has no convergent subsequence.

    (a) b_n = cos^2(a_n) (b) bn = a_n/(1 + a_n) (c) b_n = 1/(1 + |a_n|)
    \vert b_n \vert \leq 1 for (a),(b),(c) so by Bolzano-Weierstrass they have a convergent subsequence
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