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CIIM 2010 Problem 4

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June 9, 2016
CIIMCIIM 2010undergraduatefunctioncalculusderivative

Problem Statement

Let f:[0,1]→[0,1]f:[0,1] \to [0,1] a increasing continuous function, diferentiable in (0,1)(0,1) and with derivative smaller than 1 in every point. The sequence of sets A1,A2,A3,…A_1,A_2,A_3,\dots is define as: A1=f([0,1])A_1 = f([0,1]), and for n≥2,An=f(An−1).n \geq 2, A_n = f(A_{n-1}). Prove that lim⁡n→+∞d(An)=0\displaystyle \lim_{n\to+\infty} d(A_n) = 0, where d(A)d(A) is the diameter of the set AA.
Note: The diameter of a set XX is define as d(X)=sup⁡x,y∈X∣x−y∣.d(X) = \sup_{x,y\in X} |x-y|.
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