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$f$ is a continuous function from $(0,\infty) \to R$ with $f(x) \to a$ as $x \to \infty$. Can I write:

$$\lim_{X\rightarrow \infty}\lim_{n\rightarrow \infty}\sum_{i=1}^{n}f(X\cdot i/n) \cdot (1/n)= a$$

So basically, I am taking the limit on $X \to \infty$ inside the sum as it is a finite sum for a fixed $n$. That should be allowed, I believe.

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    You can't generally switch the order of two limits, and what you have written there amounts to taking $n$ out to infinity while $X$ is held fixed first, and *then* taking $X$ to infinity, though I think some analysis will show it does come out to be $a$ this way. If you try the limits the other way it's rather trivial that you will get $a$ as the limit.2012-02-16

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