Suppose $f$, $g$ are positive continuous functions on $[a,\infty)$ for some $a\in \mathbb{R}$, and $\int_a^{\infty}g(x) \,\mathrm{d}x$ diverges

Question

Problem Statement:

Suppose $f$ and $g$ are positive, continuous functions defined on $[a,\infty)$ for some $a \in \mathbb{R}$, and suppose $\int_a^{\infty}$ diverges (as an improper Riemann Integral). Show that at least one of the integrals

$$ \int_a^{\infty}f(x)g(x)dx, \hspace{10mm} \int_a^{\infty} \dfrac{g(x)}{f(x)}dx$$

diverges.

My comments:

Firstly, this is an old qualifying exam question for a Phd level real analysis where we have studied general Lebesgue Theory. For this problem I am thinking to assume that the first integral diverges, and show the other must converge. But besides that, I don't have any ideas about where to go from here. I would be happy with just a hint, and not a complete answer. Thanks!

Answer 1

\begin{align*} \int_{a}^{\infty}g(x)dx&=\int_{a}^{\infty}\sqrt{f(x)g(x)}\cdot\dfrac{\sqrt{g(x)}}{\sqrt{f(x)}}dx\\ &\leq\left(\int_{a}^{\infty}f(x)g(x)dx\right)^{1/2}\left(\int_{a}^{\infty}\dfrac{g(x)}{f(x)}dx\right)^{1/2}. \end{align*}