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Let $f,g:[a,+infty]tomathbbR$ be continuous and $K>0$ s.t. $$
left|int_c^df(x),dxright|leq K quad forall c,din [a,infty).
$$
If $gin C^1$ is decreasing with $displaystylelim_xto inftyg(x)=0,$ prove that the limit $$int_a^infty f(x)g(x),dx=lim_xtoinfty int_a^x f(t)g(t),dt$$ exists.

I tried to use the Cauchy criterion, that is $forall epsilon>0, exists A>a$ s.t. $A<c<d$ implies that $left|int_c^d f(t)g(t),dtright|<epsilon.$ However I could not get anywhere.

I appreciate any suggestions.

P.S. I already posted this question and as its expression was not completely right, I deleted it and asked it again here.

For $x in [a,infty)$, define $F(x) := int^x_a f(t),dt$. Then $lvert F(x) rvert le K$, and $F'(x) = f(x)$ by the fundamental theorem of calculus.

Now for $x,y in[a,infty)$, and wlog let $y ge x$. By integrating by parts and using the triangle inequality, we see beginalign*leftlvert int^y_a f(t) g(t) ,dt - int^x_a f(t)g(t),dtrightrvert &= left lvertint^y_x f(t) g(t), dt right rvert\
&= left lvert int^y_x F'(t)g(t) ,dt right rvert\
&= left lvert [F(t)g(t)]^t=y_t=x - int^y_x F(x)g'(x),dx right rvert\
&le lvert F(y)g(y)rvert + lvert F(x)g(x) rvert + int^y_x lvert F(t)g'(t) rvert ,dt\
& le Kleft(lvert g(y) rvert + lvert g(x) rvert + int^y_x lvert g'(t)rvert ,dt right).
endalign* Since $g$ is decreasing, we have $g'(t) le 0$, and so $lvert g'(t)rvert = -g'(t).$ Thus beginalign*leftlvert int^y_a f(t) g(t) ,dt - int^x_a f(t)g(t),dtrightrvert & le Kleft(lvert g(y) rvert + lvert g(x) rvert + int^y_x lvert g'(t)rvert ,dt right)\
&= Kleft(lvert g(y) rvert + lvert g(x) rvert - int^y_x g'(t) ,dt right)\
&= K(lvert g(y) rvert + vert g(x) rvert + g(x) - g(y)) \&le 4K textmax(lvert g(x) rvert, lvert g(y) rvert).
endalign* Since $lvert g(t) rvert to 0$ as $t to infty$, for any $epsilon > 0$, there is $M > 0$, such that $lvert g(t)rvert < epsilon/4K$ for all $t > M$. But then for $x,y > M$, we have shown $$leftlvert int^y_a f(t) g(t), dt - int^x_a f(t)g(t),dtrightrvert < epsilon.$$ Finally, take any sequence $x_n_n=1^infty$ with $x_n to infty$, and this shows that $$leftint^x_n_a f(t)g(t),dt right_n=1^infty$$ is a Cauchy sequence and hence converges. By the sequential criterion theorem, we conclude that $$lim_xtoinfty int^x_a f(t) g(t) ,dt$$ converges.

As an aside, this is completely analogous to the Dirichlet Test for infinite series, which is a generalization of the alternating series test.

Since it is given that $g$ is continuously differentiable, so $g'$ exists.
Now the idea is to write $displaystyleint fg$ as $displaystyle gint f-int left(g'int fright)$.

Now using triangle inequality,
$$left|int fgright|leqleft| gint fright|+left|int left(g'int fright)right|,$$
now I think you can complete it, but don't forget to use the fact that $displaystylelim_xtoinfty g'(x)=0.$