KöMaL 2018 A739

Let $a_1,a_2,\ldots$ be a sequence of real numbers from the interval $[0,1]$. Prove that there is a sequence $1\le n_1<n_2<\ldots$ of positive integers such that $A=\lim_{\substack{i, j\to \infty\\i\ne j}} a_{n_i+n_j}$ exists, i.e., for every real number $\varepsilon>0$ there is a $N_\varepsilon$ such that $\big|a_{n_i+n_j}-A\big| < \varepsilon$ is satisfied for any pair of distinct indices $i,j>N_\varepsilon$.