ILL 1985 Problem 78

The sequence $f_1, f_2, \cdots, f_n, \cdots$ of functions is defined for $x > 0$ recursively by $$f_1(x)=x , f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)$$ Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$