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About Limit Points

  1. Nov 23, 2008 #1
    In my textbook, it gives the following definition of a limit point:

    "A point x is a limit point of a sequence x_1,x_2,... if every open interval containing x also contains points x_n for infinitely many n"

    I am taking this to mean that every open interval containing x must also contain an infinite amount of other points from the sequence (though this doesn't necessarily mean that it contains every point of the sequence).

    Is my understanding correct?
  2. jcsd
  3. Nov 23, 2008 #2


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    Yes. For example, 0 is a limit point of the sequence [itex]a_n = 1/n[/itex].
    Although 0 is not itself in the sequence, any interval [itex](-\epsilon, \epsilon)[/itex] contains infinitely many points from the sequence. For example, for [itex]\epsilon = \frac{1}{\sqrt{10}}[/itex], the interval does not contain a1, a2 and a3, but it does contain all other elements.

    Note that the sequence needn't convert, if you take
    [tex]a_{2n} = 1/n, a_{2n + 1} = 1[/tex]
    then 0 is still a limit point (and so is 1).

    If I'm not mistaken, x is a limit point of a sequence S iff S has a subsequence S' which converges (Cauchy, and all) to S'. So basically, if part of the sequence has x as a limit. It's been a while since I did analysis though :smile:
  4. Nov 24, 2008 #3
    technically, limit pts have nothing to do with sequences
    i personally think limit pt defines some special property of cercain set
  5. Nov 24, 2008 #4
    Limit points work just as well with sequences as with sets. Since the definition of a limit point does not refer at all to the order of the sequence, it's just a special case where the set is countably infinite.
  6. Nov 24, 2008 #5


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    your way of rephrasing the definition lost slightly some precision. The definition said that there are infiniutely many INDICES n for which x(n) belongs to the set, not infinitely many POINTS.

    so the sequence could have repetitions. e.g. if the sequence is the constant sequence with all x(n) = x, then the definition is satisfied as you gave it for x to be a limit point.

    In an everyday space like the real numbers, and your language "interval" implies that is your space, it will be true that if every open interval about x contains an x(n) for even one n which is different from x, then also every interval contains an infinite number of them.

    so you could say also that if every interval around x contains even one x(n) not equal to x, then x is a limit point. however if you want the constant sequence above to have x as a limit point, then you need the version you originally gave.
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