Suppose i have $a_{n}\downarrow 0$ and $\displaystyle \sum_{n=1}^{\infty}\Delta a_{n}\log n<+\infty$, where $\Delta a_{k}=a_{k}-a_{k+1}$ and $a_{n}\downarrow 0$ means $a_{n}$ is decreasing and convergent to $0$.
I want to show that $a_{n}\log n \to 0$ as $n\to\infty.$ Clearly, $a_{n}\log n\geq 0$. I need to show that $a_{n}\log n\leq 0$ but i don't know how to solve this!