International Mathematics Competition
for University Students
2017

IMC2017: Day 2, Problem 9

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.

Proposed by: Tomáš Bárta, Charles University, Prague

Hint:What is the fixed point of the recurrence?

  Solution