IMC 2017 · Problem 9

Define the sequence $f_1, f_2, \ldots \colon [0, 1) \to \mathbb{R}$ of continuously differentiable functions by the following recurrence:

$$f_1 = 1; \qquad f'_{n+1} = f_n f_{n+1} \quad \text{on } (0, 1), \quad \text{and} \quad f_{n+1}(0) = 1.$$

Show that $\lim\limits_{n \to \infty} f_n(x)$ exists for every $x \in [0, 1)$ and determine the limit function.