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Prove that there exists a real number A [ILL 1974]

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January 2, 2011
inequalities proposedinequalities

Problem Statement

Let a be a real number such that 0<a<10 < a < 1, and let nn be a positive integer. Define the sequence a0,a1,a2,…,ana_0, a_1, a_2, \ldots, a_n an recursively by a_0 = a,   a_{k+1} = a_k +\frac 1n a_k^2   \text{ for } k = 0, 1, \ldots, n - 1. Prove that there exists a real number AA, depending on aa but independent of nn, such that 0<n(A−an)<A3.0 < n(A - a_n) < A^3.
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