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Homework Help: Is it possible to construct a sequence like this?

  1. Sep 26, 2010 #1
    1. The problem statement, all variables and given/known data
    Is it possible to construct a sequence like this? If yes, please construct one, if not, give a proof. See the attached pic for requirements for the sequence to be constructed.


    2. Relevant equations
    NO


    3. The attempt at a solution
    I know how to construct a sequence that has subsequences that converge to a finite number of limits, such as:
    1, -1, 1, -1, ...
    1, 2, 3, 1, 2, 3, ...

    This problem brings the sine function to my mind, but since R is uncountable, it's not possible to build a sequence that contains terms that has a 1-1 correspondence with sin(x).

    I might as well ask: what is the cardinality of [tex]N \times N [/tex]?
    [tex] card N \times N = card N ? [/tex]
    In that case I could just repeat the sequence {1, 1/2, 1/3, ...} over and over again.
     

    Attached Files:

  2. jcsd
  3. Sep 26, 2010 #2
    I don't know what you are referring to when you talk about the sine function considering the sequence in question only takes of rational numbers; which are countable.

    Anyway, NxN has the same cardinality has N.
     
  4. Sep 26, 2010 #3

    Office_Shredder

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    You can't just stick it after itself over and over again

    {1,1/2,1/3,...,1,1/2,1/3,....,1,1/2,1/3,..}

    isn't a sequence. Which natural number does the second 1 correspond to? You essentially need to find a bijection between N and NxN in order to write down the sequence you want
     
  5. Sep 28, 2010 #4
    If NxN has the same cardinality has N, then by definition there is a bijection between N and NxN.
     
  6. Sep 28, 2010 #5
    There are many of them.

    In fact, a one to one function on N would do.
     
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