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Two Sequences

Source: Junior Olympiad of Malaysia Shortlist 2015 A6

July 17, 2015
algebra

Problem Statement

Let (an)n≥0(a_{n})_{n\ge 0} and (bn)n≥0(b_{n})_{n\ge 0} be two sequences with arbitrary real values a0,a1,b0,b1a_0, a_1, b_0, b_1. For n≥1n\ge 1, let an+1,bn+1a_{n+1}, b_{n+1} be defined in this way: an+1=bn−1+bn2,bn+1=an−1+an2a_{n+1}=\dfrac{b_{n-1}+b_{n}}{2}, b_{n+1}=\dfrac{a_{n-1}+a_{n}}{2} Prove that for any constant c>0c>0 there exists a positive integer NN s.t. for all n>Nn>N, ∣an−bn∣<c|a_{n}-b_{n}|<c.
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