MathDB

20

Part of 1976 IMO Longlists

Problems(1)

Existence of positive number that satisfies inequality

Source:

1/3/2011
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.
inequalitiesinductionnumber theory unsolvednumber theory