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Math Help - Accumalation points.

  1. #1
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    Unhappy Accumulation points,Limit Points

    Prove that x is an accumulation point of a set
    A in a metric space X if and only if for each r > 0 the set B(x, r) \ A is infinite.

    Im findinding it hard to understand what is an accumulation point, I just dont know how to start this proof.
    Last edited by Dreamer78692; March 12th 2011 at 11:15 PM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    I suppose you meant: B(x,r)\cap A infinite.

    \Rightarrow) If B(x,r_0)\cap A=\{a_1,\ldots,a_m\} finite, choose r=\min \{d(x,a_i)\}\;(x\neq a_i) then, (B(x,r)-\{x\})\cap A=\emptyset .

    \Leftarrow) Try it.
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  3. #3
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    p is an accumulation point of a set, A, if and only if there exist other points of A arbitrarily close to p. For example, in the set (0, 1)U {2}, the open interval from 0 to 1 together with the single point 2, the set of accumulation points is precisely [0, 1]. 2 is NOT an accumulation point because there are no other points of the set within distance 1 of the point.
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  4. #4
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    Do i do something like this..

    Proof By Contradiction:

    Suppose x is an accumulation point of A, and that B(x,r)∩A is finite......

    I don't really know where to go from here, what is meant by accumulation point, I understand the concept... But just can't write it down mathematically...
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  5. #5
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    Quote Originally Posted by Dreamer78692 View Post


    Do i do something like this..

    Proof By Contradiction:

    Suppose x is an accumulation point of A, and that B(x,r)∩A is finite......

    I don't really know where to go from here, what is meant by accumulation point, I understand the concept... But just can't write it down mathematically...
    That's a good start. Now look the distance from each point in B(x,r)\cap A to x. Since there are only a finite number of points, none of which is equal to x, there must be a smallest non-zero distance. Now take a new B(x, r) with r smaller than that smallest distance.
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