Let F(x),G(x) be nonnegative decreasing functions in [0,+∞), withlim
(1) Prove that: \forall \varepsilon>0,we have \displaystyle \lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}}=0
(2) If we have \lim_{n\rightarrow+\infty}{\int_{0}^{+\infty}{(F(t)-G(t))\cos{\frac{t}{n}} dt}}=0
then prove that
\lim_{x\rightarrow0}{\int_{0}^{+\infty}{(F(t)-G(t))\cos{(xt)}dt} }=0
I tried let f(x)=\lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}} ,then for a fixed value of x,by Dirichlet test,we can see the f(x)=\lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}}
is convergence,then I have no idea about the next step.:(
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