Let $f$ be continuous on [a,b) such that $\int^b_a f$ converges. If $g'$ is locally integrable and has a constant sign on [a,b), prove that $\int^b_a fg$ converges.
Edit: We can assume that the limit of g(x) as x approaches b from the left is equal to 0.
I have been trying to use the Dirichlet's Test for Integrals here, but I am unable too meet the requirements for application. Could you help me with this one?