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Thread: Proving Theorems

  1. #1
    Oct 2008

    Proving Theorems


    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.

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  2. #2
    Oct 2008

    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 $\displaystyle p^a$ = $\displaystyle x \forall S : f(x) <= a$ 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 $\displaystyle P^a$
    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?]


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

    IF λ $\displaystyle \forall$ [0,1]

    then $\displaystyle 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)

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

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

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

    confirms $\displaystyle p^a$ 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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