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Math Help - Proving limits of integrals

  1. #1
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    Proving limits of integrals

    Problem: Define the rectangles
    R_n = [0, (n-1)/n] X  [0, (n-1)/n] = \{(x,y): 0 \leq x \leq (n-1)/n and 0 \leq y \leq (n-1)/n \}
    Also define R = [0,1] X [0,1]. Suppose f is a bounded function on R and that f is integrable over each R_n.
    (a) Prove that f is integrable over R.
    (b) Define the integrals
    I_n = \int\int _R_n f dA and I = \int\int _R f dA
    Prove that \lim n \rightarrow \infty I_n = I
    -------------------------------------------------------------------------------

    Can I get a hint on how to start (b)?
    Last edited by MissMousey; November 21st 2011 at 01:01 PM.
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  2. #2
    Super Member girdav's Avatar
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    Re: Proving limits of integrals

    Using the fact that f is bounded, show that the area of R\setminus R_n tends to 0.
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  3. #3
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    Re: Proving limits of integrals

    Quote Originally Posted by girdav View Post
    Using the fact that f is bounded, show that the area of R\setminus R_n tends to 0.
    I was told that this might not work because the area of R\setminus R_n might not be of zero content.
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