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Analysis Help: Limit Points

  1. Oct 24, 2011 #1
    Hi, I am asked to find the limit points of the given set A

    A={((-1)^n)(2n/(n-4))}

    so far I think 0 is a limit point, but I can't figure out if there are any more than just that one.

    I believe every point in A is an isolated point, and that this set is open because every point in A contains a neighborhood contained in the set.

    Any suggestions would be helpful. Thanks!!
     
  2. jcsd
  3. Oct 24, 2011 #2

    pwsnafu

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    Why do you think that?
    Consider |An|. What happens as n goes to infinity?
    If a point is isolated, how can there be an open neighborhood?
     
  4. Oct 25, 2011 #3
    I guess I thought the limit of infinity/infinity would be 0, but now that I look at it more I feel like the limit would be 2, and because we have the (-1) in front, would the limits be -2 and 2?
     
  5. Oct 25, 2011 #4

    HallsofIvy

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    You are using the word "limit" very carelessly. The sequence 2n/(n-4) has limit 2. That means its "set of limit points" is {2}. The sequence [itex](-1)^n2n/(n-4)[/itex] does not converge- it has no limit. It does have two convergent subsequences- taking only even indices, it converges to 2, taking only odd indicees it converges to -2. So its "set of limits points" (not "its limits") is {-2, 2}.
     
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