Let $f: \Bbb R \to \Bbb R $ be a monotonic decreasing function such that $\displaystyle{\lim_{x\to \infty}} f(x)= 0$. Prove that:
$$\lim_{n\to \infty} n \sum_{j=1}^n \frac{\cos\big(\frac{n}{j}\big)f\big(\frac{n}{j}\big)}{j^2} < \infty$$
My only thought about it was maybe using the integral test for convergence, but I didn't find a way to do it.