If $\int_a^{\infty} f(x)\, dx$ converges absolutely and $g(x)$ is continuous and bounded on $[a, \infty)$, then $\int_0^{\infty} f(x)g(x)\, dx$ converges.
Is this statement true or false?
I know that this statement is false if $\int_a^{\infty} f(x)\, dx$ merely converges—take $f(x) = \frac{\sin{x}}{\sqrt{x}}$ and $g(x) = \sin{x}$. However, I believe this statement is true for absolute convergence. Is the following proof correct? Thank you.
Proof. Let $\int_a^{\infty} f(x)\, dx$ converge absolutely and $g(x)$ be continuous and bounded on $[a,\infty)$. Let $M\in \mathbb{R}$ be such that $|g(x)|\le M$ for all $x\in [a, \infty)$. Then, $x\mapsto |f(x)\, g(x)|$ and $x\mapsto |M\, f(x)|$ are continuous on any closed interval $[a,b]\subset [a, \infty)$, and $$ 0\le |f(x)\, g(x)|\le |M f(x)| $$ on $[a,\infty)$. So, by the comparison test, since $$ \int_a^{\infty} |M f(x)|\, dx = |M| \int_a^{\infty} |f(x)|\, dx \;\mbox{ converges} ,$$ we have that $$ \int_a^{\infty} |f(x)\, g(x)|\, dx \;\mbox{ converges} .$$