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Define the sequence

f_1,f_2,\ldots :[0,1)\to \mathbb{R}

of continuously differentiable functions by the following recurrence: f_1=1; \qquad 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.

9

Problems(1)