Putnam 1983 B5

Let $\lVert u\rVert$ denote the distance from the real number $u$ to the nearest integer. For positive integers $n$ , let $$a_n=\frac1n\int^n_1\left\lVert\frac nx\right\rVert dx.$$ Determine $\lim_{n\to\infty}a_n$ .