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Existence of positive number that satisfies inequality

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January 3, 2011
inequalitiesinductionnumber theory unsolvednumber theory

Problem Statement

Let (an),n=0,1,...,(a_n), n = 0, 1, . . ., be a sequence of real numbers such that a0=0a_0 = 0 and an+13=12an2−1,n=0,1,⋯a^3_{n+1} = \frac{1}{2} a^2_n -1, n= 0, 1,\cdots Prove that there exists a positive number q,q<1q, q < 1, such that for all n=1,2,…,n = 1, 2, \ldots , ∣an+1−an∣≤q∣an−an−1∣,|a_{n+1} - a_n| \leq q|a_n - a_{n-1}|, and give one such qq explicitly.
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