A. 739

Part of KoMaL A Problems 2018/2019

Problems(1)

Limit of a_{n_i+n_j}

Source: KöMaL A. 739

a_1,a_2,\dotsc

be a sequence of real numbers from the interval

[0,1]

. Prove that there is a sequence

1\leqslant n_1<n_2<\dotsc

of positive integers such that

A=\lim_{\substack{i,j\to \infty \\ i\neq j}} a_{n_i+n_j}

exists, i.e., for every real number

\epsilon >0

, there is a constant

N_{\epsilon}

|a_{n_i+n_j}-A|<\epsilon

is satisfied for any pair of distinct indices

i,j>N_{\epsilon}