Let $f, g: [a, \infty) \rightarrow \mathbb{R}$ be real functions s.t $f$ is continues at $[a, \infty)$ and $\lim_{x\to\infty}f(x) = 0$.
If $g$ is Riemann-integrable at $[a, N] \space\space, \forall N \in (a, \infty)$ and $G: [a, \infty) \rightarrow \mathbb{R}$ given by, $$ \begin{array}{c} G(x) = \int_{a}^{x} |g(\xi)| \space\space d{\xi} \end{array} $$ is bounded, then $$ \begin{array}{c} \int^{\infty}_{a} fg(\xi)\space\space d{\xi} \end{array} $$
converges absolutely.
Now if we let some $\alpha_1, \alpha_2 \in [a, \infty)$, then,
$\int_{\alpha_1}^{\alpha_2}|fg(x)|\space dx = \int_{\alpha_1}^{\alpha_2}|f(x)|\cdot|g(x)|\space dx$
I'm thinking that if I'd be able to equivalently express $\int_{\alpha_1}^{\alpha_2}|fg(x)|\space dx$ in terms of $G$ and $|f|$, it's done (by using the other assumptions and Cauchy's equivalence). The only idea I've came up with so far is using the MVT theorem for integrals, which I struggle to reason why I can do that.
I'd like to have some guidance. Thanks
Hints:
(1) The function $f$ is bounded on $[a,\infty)$.
(2) The function $G$ is bounded by hypothesis and non-decreasing, so $\int_a^\infty |g(\xi)| \, d\xi = \lim_{x \to \infty}\int_a^x |g(\xi)| \, d\xi$ exists.
Now show that the Cauchy criterion for the improper integral of $fg$ is satisfied.