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Math Help - SupA.SupB=InfAB

  1. #1
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    Smile SupA.SupB=InfAB

    Querrian ayudarme con lo siguiente:

    A,B, subconjuntos de R acotados y no vacíos

    a)Si SupA<0 y SupB<0
    Demostrar:
    Inf(AB)=SupA.SupB
    gracias
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  2. #2
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    You probably want to show:

    \inf AB \leq \sup A * \sup B and,
    \inf AB \geq \sup A * \sup B.

    Hint: (contradicción).

    Hmm.. Maybe I can get you started, one of the directions seems simple.

    a \leq \sup A, \forall a \in A.
    b \leq \sup B, \forall b \in B.

    Now,

    a*b \leq \sup A * \sup B. [MISTAKE?]

    Also,

    \inf AB \leq a*b, \forall a \in A, and b \in B.

    So,

    \inf AB \leq a*b \leq \sup A * \sup B, and thus

    \inf AB \leq \sup A * \sup B
    Last edited by cgiulz; October 1st 2009 at 12:28 PM.
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  3. #3
    Junior Member qspeechc's Avatar
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    Quote Originally Posted by cgiulz View Post
    a*b \leq \sup A * \sup B.
    Hmm, I dunno bout this:

    -5<-3 and -2<0 but (-5)*(-2)>(-3)*0

    I don't speak the language of the original post, but I think it translates as this in English:

    If A and B are non-empty subset of real numbers such that
    supA< 0 and supB< 0
    show that:
    infAB = supA.supB

    Here's what I would do:

    Since supA< 0 it means there is a negative number w such that a\leq w<0 for all a in A. Also, there is a v<0 such that b\leq v<0 for all b in B.
    So ab\geq vw for all a in A and b in B, which we can do because of the signs of everything. This shows that vw is a lower bound for AB, i.e. supA.supB is a lower bound for AB. All that remains to show is that vw is the largest lower bound. Try to see if you can show this.
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  4. #4
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    I think I was assuming the converse.
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  5. #5
    Junior Member qspeechc's Avatar
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    If someone could translate the original post to English it would be very helpful
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  6. #6
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    Quote Originally Posted by MATHFRIEND View Post
    Querrian ayudarme con lo siguiente:

    A,B, subconjuntos de R acotados y no vacíos

    a)Si SupA<0 y SupB<0
    Demostrar:
    Inf(AB)=SupA.SupB
    gracias
    Woud you like to help me with the following:

    A,B subsets of R, bounded and non-empty.

    a)If SupA<0 and SupB<0
    Prove:
    Inf(AB)=SupA.SupB
    Thanks
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