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How do you prove that a finite set of points cannot have any accumulation points in a complex analysis?
Wiki User
∙ 11y ago
Complex analysis is a metric space so neighborhoods can be described as open balls. Proof follows
a. Assume that the set has an accumulation point call it P.
b. An accumulation point is defined as a point in which every neighborhood (open ball) around P contains a point in the set other than P.
c. Since P is an accumulation point, I can choose an open ball around P that has a diameter less than the minimum distance between P and all elements of the finite set. Therefore there exists a neighbor hood around P which contains only P. Therefore P is not an accumulation point.
Wiki User
∙ 11y ago
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How do you prove that a finite set of points cannot have any accumulation points in a real analysis?
If you have a finite set of points (call them A1, A2, A3...), then you have a finite set of distances to the points. So for any point B, simply pick a distance D that's smaller than the distance between B and A1, the distance between B and A2, and so on. (This is possible, since there a finite number of points.) ================================================ Since there are no points within distance D of B (because this is how you chose D), point B can not be an accumulation point (because an accumulation point must have points within any given distance of it)
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How do you prove that a finite set of points cannot have any accumulation points in a real analysis?
If you have a finite set of points (call them A1, A2, A3...), then you have a finite set of distances to the points. So for any point B, simply pick a distance D that's smaller than the distance between B and A1, the distance between B and A2, and so on. (This is possible, since there a finite number of points.) ================================================ Since there are no points within distance D of B (because this is how you chose D), point B can not be an accumulation point (because an accumulation point must have points within any given distance of it)
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