$\lim_{n\to\infty} \sum\limits_{k=1}^n \frac{\ln k}{n} ( 1-\{\frac{n}{k} \} ) ( (1-\frac{k}{n} \{\frac{n}{k}\} )-\frac{1}{2})$

Question

Can someone help me prove that the limit approaches zero, I know it does, but I can't prove it.

$$\lim_{n\to\infty} \sum\limits_{k=1}^n \frac{\ln k}{n} \left( 1-\left\{\frac{n}{k} \right\} \right) \left( \left(1-\frac{k}{n} \left\{\frac{n}{k}\right\} \right)-\frac{1}{2} \right)$$

where $\displaystyle \left\{ \frac{n}{k} \right\}$ is the fractional part of $\displaystyle \frac{n}{k}$.