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Math Help - Help with proof (sets)

  1. #1
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    Help with proof (sets)

    Just wondering if someone could help me with this problem.

    For two disjoint compact subsets of R^n, show that there is a minimum positive distance between the points of each set.

    Much appreciated!
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  2. #2
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    Quote Originally Posted by gasbasis View Post
    Just wondering if someone could help me with this problem.
    For two disjoint compact subsets of R^n, show that there is a minimum positive distance between the points of each set.
    Here is an outline.
    Suppose that the two subsets are A & B and that D(A,B)=0.
    Use the definition of distance between sets to get a sequence of distinct points in A that must converge to a point in B. That happens due to compactness. That will give you a contradiction.
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  3. #3
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    We can generalize this. If A is compact and B is closed subsets of R^n then d(A,B) > 0
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  4. #4
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    Quote Originally Posted by Plato View Post
    Here is an outline.
    Suppose that the two subsets are A & B and that D(A,B)=0.
    Use the definition of distance between sets to get a sequence of distinct points in A that must converge to a point in B. That happens due to compactness. That will give you a contradiction.
    Thanks a lot for this. Does it make sense to also say that since A,B are compact, the function that describes a path between a point in A and a point in B must have a minimum and a maximum because it is a function on a compact set?
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