IMO 1985 Problem 6

For every real number $x_1$ , construct the sequence $x_1,x_2,\ldots$ by setting: $$x_{n+1}=x_n(x_n+{1\over n}).$$ Prove that there exists exactly one value of $x_1$ which gives $0<x_n<x_{n+1}<1$ for all $n$ .