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Limit of a_{n_i+n_j}

Source: KöMaL A. 739

January 21, 2019
algebra

Problem Statement

Let a1,a2,…a_1,a_2,\dotsc be a sequence of real numbers from the interval [0,1][0,1]. Prove that there is a sequence 1⩽n1<n2<…1\leqslant n_1<n_2<\dotsc of positive integers such that A=lim⁡i,j→∞i≠jani+njA=\lim_{\substack{i,j\to \infty \\ i\neq j}} a_{n_i+n_j}exists, i.e., for every real number ϵ>0\epsilon >0, there is a constant NϵN_{\epsilon} that ∣ani+nj−A∣<ϵ|a_{n_i+n_j}-A|<\epsilon is satisfied for any pair of distinct indices i,j>Nϵi,j>N_{\epsilon}.
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