### International Mathematics Competition for University Students

July 31 – August 6 2017, Blagoevgrad, Bulgaria

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