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LUB axiom related question

  1. Aug 8, 2011 #1
    1. The problem statement, all variables and given/known data
    h: X x Y --> R is a function from X x Y to R. X,Y nonempty. If range is bounded in R. then let

    f : X --> R st f(x) = sup{h(x,y): y belongs to Y} (call this set A)
    g :Y --> R st g(y) = inf{h(x,y) : x belongs to X} (call this set B)

    Then prove that

    sup{g(y) : y belongs to Y} is less than or equal to inf{f(x) : x belongs to X}


    2. Relevant equations

    none.
    3. The attempt at a solution

    As h(X,Y) is bounded the LUB and GLB exist. Now for each x, A is a subset of h(x,y). thus f(x) is <= LUB.
    thus inf f(x) <= LUB.

    Similarly I got, sup g(y) >= GLB.

    But this leaves me nowhere :(
     
    Last edited: Aug 8, 2011
  2. jcsd
  3. Aug 8, 2011 #2
    well Ive got it, thanks anyways.
     
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