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Math Help - convering subsequences

  1. #1
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    convering subsequences

    Provide an example of a sequence with a given property:

    (a) a sequence that has subsequences that converge to 1, 2, and 3.
    (b) a sequence that has subsequences that converge to infinity and negative infinity.
    (c) a sequence that has a strictly increasing subsequence, a strictly descreasing subsequence, and a constant subsequence.
    (d) an unbounded sequence which has a convergent subsequence
    (e) a sequence that has no convergent subsequence

    Thanks for any help!
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  2. #2
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    Quote Originally Posted by friday616 View Post
    Provide an example of a sequence with a given property:
    (a) a sequence that has subsequences that converge to 1, 2, and 3.
    I will give you example for this one.
    You try the others.
    x_n  = \bmod (n,3) + 1 + \frac{1}{n}
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  3. #3
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    For (e) does \pm\infty count as limits? Because if they do and \limsup and \liminf always exist in the extended reals, the problem is impossible (in \mathbb{R} at least)

    If not then use a_n=n
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