Suppose we have a sum $$\sum_{n<x}\frac{f(n)g(n)}{n},$$ and we know asymptotically the value of $\sum_{n<x}\frac{f(n)}{n}\sim h(x)$, and that $g(x)=O(j(x))$ for another function $j$. Is it then valid to conclude that $$\sum_{n<x}\frac{f(n)g(n)}{n}=O(h(x)j(x))?$$
To me it seems as though this should intuitively be true, but cannot find a proof anywhere. The intuitive idea I have is that $g(n)$ is bounded so we should be able to use some result like monotone convergence/dominated convergence. But I have no idea what the correct proof/counterexample is.