OBM 1990 Problem 5

Let $f(x)=\frac{ax+b}{cx+d}$ $F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$ )

Suppose that $f(0) \not =0$ , $f(f(0)) \not = 0$ , and for some $n$ we have $F_n(0)=0$ , show that $F_n(x)=x$ (for any valid x).