# Thread: Proving limits of integrals

1. ## Proving limits of integrals

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

2. ## Re: Proving limits of integrals

Using the fact that $\displaystyle f$ is bounded, show that the area of $\displaystyle R\setminus R_n$ tends to $\displaystyle 0$.

3. ## Re: Proving limits of integrals

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