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Homework Help: Subsequential limit point

  1. Aug 27, 2012 #1
    1. The problem statement, all variables and given/known data

    Determine all subsequential limit points of the sequence X_n = cos(n)

    2. Relevant equations

    Unsure of any.

    3. The attempt at a solution

    Tried determining subsequences of cos(n) but, having trouble finding any.

    Can anyone tell me the definition and how to proceed?

  2. jcsd
  3. Aug 27, 2012 #2


    Staff: Mentor

    The definition of what?
  4. Aug 27, 2012 #3
    The definition of subsequential limit point.
  5. Aug 27, 2012 #4


    Staff: Mentor

    A limit point of a subsequence.

    So that it doesn't appear that I'm being flip, here's an example to illuminate this concept. Consider an = cos(n * ##\pi/2##), n ≥ 1.

    The first few terms of the sequence: {0, -1, 0, 1, 0, -1, 0, 1, ...}

    0 is a limit point of the subsequence {0, 0, 0, ... }.
    Likewise, -1 and 1 are limit points of the subsequences {-1, -1, -1, ...} and {1, 1, 1, ...}, respectively.
  6. Aug 28, 2012 #5


    User Avatar
    Science Advisor

    To give yet another perspective, consider your sequence as just a collection of

    points. A collection of points may have more than one limit point ( or, of course,

    no limit points or exactly one limit point). Informally, a subsequence is a subset of the

    original sequence in which the order of the terms is preserved.
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