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Prove that there exist one number a

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August 28, 2010
functionrecurrence relationIMO 1985IMO ShortlistFixed pointcalculuslimit

Problem Statement

The sequence f1,f2,⋯ ,fn,⋯f_1, f_2, \cdots, f_n, \cdots of functions is defined for x>0x > 0 recursively by f_1(x)=x ,   f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right) Prove that there exists one and only one positive number aa such that 0<fn(a)<fn+1(a)<10 < f_n(a) < f_{n+1}(a) < 1 for all integers n≥1.n \geq 1.
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