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Math Help - Principles of Mathematical Analysis - Ch2 Basic Topology

  1. #1
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    Principles of Mathematical Analysis - Ch2 Basic Topology

    Hi,

    Here I am reading Rudin's Book(3rd).
    I encountered some problems in Theorem 2.20, page 32.

    Theorem 2.20
    If p is a limit point of a set E, then every neighborhood of p contains infinitely many points of E.

    By intutition or imagination, I can understand the theorem, but I just don't understand why it says,

    "The neighborhood Nr(p) contains no point q of E such that p=/=q (not equal), so that p is not a limit point of E." in the proof.

    I think it would be that Nr(p) contains some q when r is in some range, but in other range of r, there is no q of E such that p=/=q. In other words, not every neighborhood of p contains a q of E, such that p=/=q.

    Is there anyone who has the book on hands?
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  2. #2
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    So he has supposed there is only a finite number of points in a neighborhood around a limit point of the set, and basically taken a smaller neighborhood such that those points aren't there anymore. Since we don't know a priori whether or not p is an element of E, the statement you are questioning is simply stating we exhibit a neighborhood of p with no elements of E other than possibly p, which contradicts the definition of a limit point.
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  3. #3
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    If there are only a finite number of points of E in a neighborhood of p, the there is a closest such point. Take \epsilon to be half that distance.
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  4. #4
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    Quote Originally Posted by siclar View Post
    So he has supposed there is only a finite number of points in a neighborhood around a limit point of the set, andbasically taken a smaller neighborhood such that those points aren't there anymore. Since we don't know a priori whether or not p is an element of E, the statement you are questioning is simply stating we exhibit a neighborhood of p with no elements of E other than possibly p, which contradicts the definition of a limit point.
    So he took another r' smaller than the min(r)? Because by the text I couldn't tell. As for the statement, it was obviously volative of the definition of a limit point. I have no idea why I made it I. I guess I was just so confused by the r.

    Thank you very much! I've known what my problem is!
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  5. #5
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    Quote Originally Posted by HallsofIvy View Post
    If there are only a finite number of points of E in a neighborhood of p, the there is a closest such point. Take \epsilon to be half that distance.
    Thank you very much. I got it!
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