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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Originally Posted by cgiulz

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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for (a),(b),(c) so by Bolzano-Weierstrass they have a convergent subsequence