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Prove: $\int_{0}^{\infty} \sin (x^2) dx$ converges.

I have shown that; $\forall \epsilon>0, \exists r\in \mathbb{R}$ such that $ \forall x,y>0, r.

Also, i have shown that $\forall x,y>0, |\int_{x}^{y} \sin (t^2) \, dt| < 1/x$.

How do i prove that $\lim_{x\to\infty} \int_0^x \sin (t^2) \, dt$ converges?

EDIT:

Please do not close this post. I saw Davide's answer in the link, but don't understand his argument. Why does convergence of $\int_{0}^{\infty} t^{-3/2} dt$ imply that $\int_{a}^{\infty} t^{-3/2} \cos t dt$ converges?

I think that only implies that limsup and liminf of $\int_{a}^{\infty} t^{-3/2} \cos t dt$ is finite. And that's exactly what i said at the first of the sentence in my post.

Of course, i tried to convert this integral to a limit of a series, to apply 'alternating series test'. So i was trying to prove a lemma, but i failed to prove and it is indeed false. (Check this in the comment below) (To be specific, i was trying to show that $\int_{0}^{\infty} \sin t^2 dt = \sum_{n=0}^{\infty} \int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}} \sin t^2 dt$)

To summarize, what is a theorem that is a generalization of that in Michael's post for Riemann Stieltjes Integral?

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    See http://math.stackexchange.com/questions/105107/prove-int-0-infty-sin-x2-dx-converges .2012-12-09
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    The easiest way is just to use the alternating series test (some details need to be filled in).2012-12-09
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    @Potato I was trying to prove that "For any strictly increasing sequence $\{c_n\}$ in $\mathbb{R}$ such that $c_1=a$, $\int_{a}^{\infty} f d\alpha$ converges iff $\sum_{n=1}^{\infty} \int_{c_n}^{c_{n+1}} f d\alpha$ converges and their limits are the same", but i thought this is false. I have no idea how to change this limit to limit of series.2012-12-09
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    Incidentally, do you know that not only is this integral convergent, but that its exact value is $\sqrt{\pi/8}$? You can relate this to the Gaussian integral of $e^{-x^2}$ through the relation $e^{ix}=\cos(x)+i\sin(x)$. See [http://en.wikipedia.org/wiki/Fresnel_integral](http://en.wikipedia.org/wiki/Fresnel_integral).2012-12-09

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