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Math Help - Proving Theorems

  1. #1
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    Proving Theorems

    Hello,

    Can someone help me prove Theorem 6 at this page

    Alternatively there are more theorems in the book Further Mathematics for economics analysis, page 54, chapter 2 Theorem 2.4.2

    There is a little explanation on the theory but it doesn't explain too much detail.

    Thanks
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  2. #2
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    Re: Proving Theorems

    Okay this is what I have for theorem b of the book:

    I'm using the explanation for theorem a) in the book (at the bottom of page 54)

    f is convex, the set p^a =  x \forall S : f(x) <= a is convex for every number a

    I'm not fully understanding what I'm doing here but here is my answer;

    let x and y be points in P^a
    Then x and y belong to S

    NOTE: [Am i right in saying that x is the f(x) = f(xa...xn) and y is the g(x) = g(x1....xn) in this proof?]

    Continuing:

    f(x) <= a
    f(y) <= a

    IF λ \forall [0,1]

    then  λx + (1-λ)y belongs to S

    Since S is a convex set...


    As f is convex also; (I move now to the definition on page 54 of a convex set)

    \lambda x + (1- \lambda) y >= \lambda f(x) + (1- \lambda) f(y)

    for all x and y in S and all Lambda in [0,1]

    Shows that \lambda x + (1- \lambda) y in all  p^a

    confirms  p^a is convex.....

    Is this correct?

    Honestly I just changed the words from the proof of part a on page 54, and don't really understand the Theorem

    Any help would be much appreciated!
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