Question 6
The sequence $f_1,f_2,\ldots,f_n,\ldots$ of functions is defined for $x>0$ recursively by
$$f_1(x)=x,\qquad f_{n+1}(x)=f_n(x)\left(f_n(x)+\frac1n\right).$$
Prove that there exists one and only one positive number $a$ such that $0\lt f_n(a)\lt f_{n+1}(a)\lt 1$ for all integers $n\ge1$.
$$f_1(x)=x,\qquad f_{n+1}(x)=f_n(x)\left(f_n(x)+\frac1n\right).$$
Prove that there exists one and only one positive number $a$ such that $0\lt f_n(a)\lt f_{n+1}(a)\lt 1$ for all integers $n\ge1$.