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Math Help - sup/inf

  1. #1
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    sup/inf

    Hi,
    I understand the basic definition of sup and inf, but when it comes to proving the sup or inf of an infinite set where infS/supS is not an element of S, I am having trouble.

    For example, given a problem of proving that

    supS=0 where S={x|x<0},

    I am able to prove the first part of the definition of supS (proving that 0 is an upper bound of S), but I always get stuck at the 2nd part of the definition

    (proving that for any b, where b is an upper bound of S, 0 is less than or equal to b).

    Any help would be appreciated
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by lexietx View Post
    Hi,
    I understand the basic definition of sup and inf, but when it comes to proving the sup or inf of an infinite set where infS/supS is not an element of S, I am having trouble.

    For example, given a problem of proving that

    supS=0 where S={x|x<0},

    I am able to prove the first part of the definition of supS (proving that 0 is an upper bound of S), but I always get stuck at the 2nd part of the definition

    (proving that for any b, where b is an upper bound of S, 0 is less than or equal to b).

    Any help would be appreciated
    Suppose that \alpha is a lower bound for S then \forall x>0,\alpha<x\implies \alpha\leqslant 0 ?
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  3. #3
    Senior Member Pinkk's Avatar
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    Uptown Manhattan, NY, USA
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    Assume to the contrary that for there exists some bound b, 0 > b. Well, if b is a bound, then we know for b>= all the elements of the set. But then 0 is a greater bound than the bound b, so 0 cannot be the least upper bound, so Sup =/= 0, which is a contraction.
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