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Inequality on eventually constant sequence

Source: 2022 Taiwan TST Round 1 Independent Study 2-A

July 6, 2022
inequalitiesTaiwan

Problem Statement

Let a1,a2,a3,…a_1, a_2, a_3, \ldots be a sequence of reals such that there exists N∈NN\in\mathbb{N} so that an=1a_n=1 for all n≥Nn\geq N, and for all n≥2n\geq 2 we have an≤an−1+2−na2n.a_{n}\leq a_{n-1}+2^{-n}a_{2n}. Show that ak>1−2−ka_k>1-2^{-k} for all k∈Nk\in\mathbb{N}.
Proposed by usjl
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