Simplified version of the Fourier inversion formula I'm working out of Stein's Fourier Analysis (working with Riemann integrable functions), and I'm having trouble with problem 1:


Suppose $f$ is continuous and supported on $[-M,M] \subset \mathbb{R}$ such that the Fourier transform satisfies $|\hat{f}(\xi)| \leq \frac{B}{1+\xi^2}$ (of moderate decrease).
Show that:
a) Let $L$ be such that $\frac{L}{2}$ and $\delta = \frac{1}{L} > M$ 
Show $f= \delta \sum \hat{f}(\delta n)e^{-2\pi i n \delta x}$ where $\hat{f}(\delta n)$ are the Fourier coefficients


I don't have a problem with this, I used the fact that $\hat{f}$ is of moderate decrease to show that the series converges absolutely thus it converges uniformly to $f$
Now, my problem is with the part


b) Prove that if $F$ is continuous and of moderate decrease then 
$\int^{\infty}_{-\infty} F({\xi}) d\xi = \operatorname{lim}_{\delta \to 0, \delta > 0}  \delta \sum F(n\delta) $


where all sums are taken from $-\infty$ to $\infty$
The book suggests approximating the integral by $\int^{N}_{-N} F$ and the series by $\delta \sum_{|n| \leq N/\delta} F(n\delta) $ then approximating the second integral by Riemann sums.
It isn't hard to see that for $ \epsilon > 0$  $\int_{|n| > N} F < \epsilon$ as $F$ is of moderate decrease
However, I'm stuck with what I should be doing now.  Any help is appreciated.
The motivation here is that we want the Fourier inversion formula as a conclusion from the above two parts.
 A: The sum is already a Riemann sum, I think the catch is just to formalize the infinite integration domain. There isn't much of anything to do beyond what's obvious. An integral like that is defined as a limit:
$$\int_{-\infty}^\infty=\lim_{a,b\to \infty}\int_{-a}^b$$
You have to do this here as well:
$$\lim_{a,b\to\infty}\lim_{\delta\to 0^+}\delta \left(\sum_{n=0}^{b/\delta} F(n\delta)+\sum_{n=1}^{a/\delta} F(-n\delta)\right)   $$
Notice that I split the integral in two. We aren't allowed to use the same value for lower and upper integration boundary because that would be a Cauchy principal value, not a true integral.
Now this is a finite interval and you are formally allowed to treat it as a Riemann integral:
$$\lim_{a,b\to\infty}\left(\int_0^b F(x)dx+\int_{-a}^0F(x)dx\right)   $$
Because $|F(x)|\leq \frac{1}{1+x^2}$, these integrals converge and you are allowed to take the limit.
If you exchange the limits, you get infinite sums that you can bound by $n^{-2}$ so they converge as well.
As I said, it's all the matter of strictness, but the idea is already stated in your book.
