calculus - Prove the existence of a limit : $lim limits_{xrightarrow+infty}{int_{varepsilon}^{+infty}{xF(xt)cos{t}dt}}=0$

Let $F(x),G(x)$ be nonnegative decreasing functions in $[0,+\infty)$, with$\,\displaystyle \lim_{x\rightarrow+\infty}{x(F(x)+G(x))}=0$

(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.:(