IMC 2017 Problem 9

Define the sequence $f_1,f_2,\ldots :[0,1)\to \mathbb{R}$ of continuously differentiable functions by the following recurrence: $$f_1=1; f_{n+1}'=f_nf_{n+1} \text{on $(0,1)$ }, \text{and} f_{n+1}(0)=1.$$ Show that $\lim\limits_{n\to \infty}f_n(x)$ exists for every $x\in [0,1)$ and determine the limit function.