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Math Help - Infimum theorems

  1. #1
    Member aldrincabrera's Avatar
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    Cool Infimum theorems

    good day everyone,.,i need ur help once again,.,.can u help me prove this??

    infimum = alpha iff (-infinity, alpha) intersection S = null and

    [alpha, alpha + epsilon) intersection S not equal to Null for all epsilon > 0.

    sorry for the luck of symbols used,.,.thnx
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  2. #2
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    Re: Infimum theorems

    Quote Originally Posted by aldrincabrera View Post
    good day everyone,.,i need ur help once again,.,.can u help me prove this??

    infimum = alpha iff (-infinity, alpha) intersection S = null and

    [alpha, alpha + epsilon) intersection S not equal to Null for all epsilon > 0.

    sorry for the luck of symbols used,.,.thnx


    The above seems to be pretty straighforward. What definition of "minimum of non-empty real set S" do you have?

    Tonio
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  3. #3
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    Re: Infimum theorems

    clearly S is non-empty (let ε = 1, then [α,α+1) ∩ S is non-empty, so S is non-empty).

    clearly S is also bounded below: if β < α, then for any s in S, β < s.

    (if not, then since β is in (-∞,α), s would also be in (-∞,α), contradicting the fact that (-∞,α)∩ S = ).

    so inf(S) exists. now show that both inf(S) < α and inf(S) > α violate one of your two conditions.

    (hint: if inf(S) > α, use ε < inf(S) - α).
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    Re: Infimum theorems

    Quote Originally Posted by Deveno View Post
    clearly S is non-empty (let ε = 1, then [α,α+1) ∩ S is non-empty, so S is non-empty).



    Uuh?? How can you know if S isn't given by the OP??

    Tonio





    clearly S is also bounded below: if β < α, then for any s in S, β < s.

    (if not, then since β is in (-∞,α), s would also be in (-∞,α), contradicting the fact that (-∞,α)∩ S = ).

    so inf(S) exists. now show that both inf(S) < α and inf(S) > α violate one of your two conditions.

    (hint: if inf(S) > α, use ε < inf(S) - α).
    .
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    Re: Infimum theorems

    because....(from the original post):

    "[alpha, alpha + epsilon) intersection S not equal to Null for all epsilon > 0."

    i picked an epsilon, 1. 1 > 0...what's the problem?
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