RMO 2002 Problem 6

Prove that for any natural number $n > 1$ , $$\frac{1}{2} < \frac{1}{n^2+1} + \frac{2}{n^2 +2} + \ldots + \frac{n}{n^2 + n} < \frac{1}{2} + \frac{1}{2n}.$$