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Math Help - Nested Interval

  1. #1
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    Nested Interval

    I need to prove the following:
    1.) The intersection (infinity on top, n=1 on bottom) of the open interval (0,1/n) = the null set
    2.) the intersection (infinity on top, m=1 on bottom) of the interval [m, infinity) = the null set

    How do you show these? Why don't they adhere to the Nested Interval Property?
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  2. #2
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    Quote Originally Posted by MKLyon View Post
    I need to prove the following:
    1.) The intersection (infinity on top, n=1 on bottom) of the open interval (0,1/n) = the null set
    If you had the closed intervals,
    [0,1/n]
    Then by the nested interval theorem, it would be a single point since lim 1/n---> 0
    That point would be zero.
    Since, these are open intervals that point (zero) is not included. Thus, the intersection is empty.
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  3. #3
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    Here again is another way.
    In a previous reply I gave you this lemma.
    If e>0 then is a positive integer K such that (1/K)<e.
    Looking at the intersection of the collection of sets (0,1/n), if it were not empty then it would contain some e>0. But e is not in set (0,1/K) in the collection, hence a contradiction. The intersection must be empty.
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  4. #4
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    How would you go about proving the second one:


    the intersection (infinity on top, m=1 on bottom) of the interval [m, infinity) = the null set.

    Thanks for any help.
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  5. #5
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    Well, once again if there were a number in the intersection of the [m,oo) that number would be an upper bound for the positive integers. That is a contradiction.
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