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Prove that a converging sequence is bounded

  1. Aug 31, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose that the sequence {an}converges. Show that the sequence {an} is bounded.

    3. The attempt at a solution
    Since the sequence converges, for every delta>0, there must exist a number N such that for every n>=N,
    |an - x|< delta. Therefore, for n>=N, -delta+x < an < delta + x.
    So I've proven that for n>=N, the sequence is bounded between -delta+x and delta+x.

    But I don't know how to prove that for n<N, an is also bounded. I know that there are only a finite number of elements before the sequence starts to converge. Is there a theorem stating that all finite sets are bounded?
     
  2. jcsd
  3. Aug 31, 2008 #2
    I would consider looking at the maximum of a finite set.
     
  4. Aug 31, 2008 #3

    statdad

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    Homework Helper

    To expand a little:
    Pick an [tex] \varepsilon > 0 [/tex], and use convergence to conclude that

    [tex]
    |x_i - a | < \varepsilon
    [/tex]

    for all [tex] i \ge N [/tex].

    You now have two sets of elements of the sequence: those with [tex] i \ge N [/tex] and those for smaller [tex] i [/tex]. You should be able to argue that both sets are bounded, which means ...
     
    Last edited by a moderator: Aug 31, 2008
  5. Aug 31, 2008 #4

    Dick

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    Science Advisor
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    Yes. All finite sets are bounded. Prove it by induction. (If you really need a proof).
     
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