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Convergence of a sequence of integers

  1. Mar 9, 2010 #1
    1. The problem statement, all variables and given/known data

    Given a Cauchy sequence of integers, prove that the sequence is eventually constant.

    2. Relevant Definitions and Theorems

    Definition of Cauchy sequences and convergence
    Monotone convergence
    Every convergent sequence is bounded
    Anything relevant to integers

    3. The attempt at a solution

    I can see why the theorem is true. I thought that, since the sequence in nonempty and bounded, the supremum and infimum of the set containing the sequence both exist and that the limit of the sequence must be a number between the infimum and the supremum. But I got stuck trying to prove that the limit is contained within that set (that the limit is also an integer). I don't know if it's my approach that's leading me to a dead end or if there's a theorm I've overlooked or something else entirely. Maybe I'm making it overly complicated? Any help would be appreciated.
     
  2. jcsd
  3. Mar 9, 2010 #2
    Look at the definition of Cauchy sequence. Choose epsilon<1. Conclude.
     
  4. Mar 9, 2010 #3
    Start from the definition of a Cauchy sequence.

    It usually starts "For all epsilon greater than zero, [...]"
    Can you choose an epsilon that gives your conclusion?
     
  5. Mar 9, 2010 #4

    I'm afraid I don't see what directon either of you are going in. As far as I know, the definition of a Cauchy sequence allows me to make the distance between members of the sequence small--it doesn't make it zero, which is what I need for the sequence to be constant for large n.
     
  6. Mar 9, 2010 #5

    Mark44

    Staff: Mentor

    The point Tinyboss and Mathnerdmo are making is that the sequence consists of integers.
     
  7. Mar 9, 2010 #6
    And what does that mean for my proof?
     
  8. Mar 9, 2010 #7

    Mark44

    Staff: Mentor

    It says that there is some minimum distance between elements in the sequence.
     
  9. Mar 9, 2010 #8
    Oh! That's right! Thanks so much!
     
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