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Math Help - Sequence, Supremum, Infimum

  1. #1
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    Sequence, Supremum, Infimum

    Suppose that A and B are nonempty sets of real numbers and that for every number x in A and every number y in B we have x < y. Prove that the following conditions are equivalent:
    (1) We have sup A = inf B
    (2) There exists a sequence (X sub n) in the set A and a sequence (Y sub n) in the set B such that (Y sub n)-(X sub n) approaches 0 as n approaches infinity.
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  2. #2
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    Quote Originally Posted by Slazenger3 View Post
    Suppose that A and B are nonempty sets of real numbers and that for every number x in A and every number y in B we have x < y. Prove that the following conditions are equivalent:
    (1) We have sup A = inf B
    (2) There exists a sequence (X sub n) in the set A and a sequence (Y sub n) in the set B such that (Y sub n)-(X sub n) approaches 0 as n approaches infinity.
    What do you have so far?
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  3. #3
    Senior Member Tinyboss's Avatar
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    If sup A and inf B are not equal, then they differ by some positive distance, right?
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  4. #4
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    Quote Originally Posted by Tinyboss View Post
    If sup A and inf B are not equal, then they differ by some positive distance, right?
    Yes, that's true of any two numbers! I think you also need to show that sup A\le inf B
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