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The set of convergent subsequences

  1. Feb 7, 2012 #1

    zbr

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    Hello all,

    (*) I have a question about convergent subsequences. Specifically I am looking for an example of a sequence that is unbounded but who has convergent subsequences in the interval [0,1].

    A similar question would be to have an unbounded sequence, but who has a convergent subsequence to a specific number, let's say 0.

    For this I would take the sequence:
    a_n = -1,0,1,-2,0,2,-3,0,3,...,-n,0,n

    Can I do a similar thing for (*)? i.e Can I take the following sequence:

    a_n = -1, [0,1], 1, -2, [0,1], 2,...,-n, [0,1], n

    My intuition tell me no since there will be infinite terms in each of the intervals [0,1].

    Thanks in advance for any and all help!
     
  2. jcsd
  3. Feb 7, 2012 #2

    Dick

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    You can arrange all of the rational number in [0,1] into a sequence, right? Now throw some extra numbers in to make it unbounded, just like you did for the zero sequence.
     
  4. Feb 7, 2012 #3

    zbr

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    Ah, yes of coarse, set sequence 1, 1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, 1/16, ... would satisfy the sequence with convergent subsequences within the interval [0,1], then just throwing in divergent terms would satisfy the second requirement.

    Thanks so much for your help Dick!
     
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