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Math Help - Tough Problem, need suggestions.

  1. #1
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    Tough Problem, need suggestions.

    Let f,g: [0,1] → ℝ be two continuos functions. Prove that

    lim ∫{x=0,x=1} f(x^n)g(x)dx = f(0) ∫{x=0,x=1} g(x)dx.
    n→∞

    The things inside the brackets are the limits of the integrals
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by auitto View Post
    Let f,g: [0,1] → ℝ be two continuos functions. Prove that

    lim ∫{x=0,x=1} f(x^n)g(x)dx = f(0) ∫{x=0,x=1} g(x)dx.
    n→∞

    The things inside the brackets are the limits of the integrals
    Just prove that
    \left|\int_0^1 f(x^n)\cdot g(x)\, dx-f(0)\cdot\int_0^1 g(x)\, dx\right|\rightarrow 0, \quad \text{ as } n\rightarrow \infty
    or, maybe better still, that
    \left|\int_0^1 \Big(f(x^n)-f(0)\Big)\cdot g(x)\, dx\right|\rightarrow 0, \quad \text{ as } n\rightarrow \infty

    Because, consider:
    (1) f being continuous implies that for any \varepsilon>0 there exists a \delta >0 such that |f(x)-f(0)|<\varepsilon if 0\leq x< \delta.
    (2) the larger n the longer does x^n stay close to 0 (for example, smaller than \delta) as you move x from 0 to 1.
    (3) f and g being continuous in [0;1] also implies that |f(x^n)\cdot g(x)|\leq M for some constant M and all x\in [0;1].
    Last edited by Failure; April 8th 2010 at 08:42 AM.
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  3. #3
    Math Engineering Student
    Krizalid's Avatar
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    you can also invoke the dominated convergence theorem but i don't know if you've covered yet.
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