CIIM 2018 Problem 6

Let $\{x_n\}$ be a sequence of real numbers in the interval $[0,1)$ . Prove that there exists a sequence $1 < n_1 < n_2 < n_3 < \cdots$ of positive integers such that the following limit exists $$\lim_{i,j \to \infty} x_{n_i+n_j}.$$ That is, there exists a real number $L$ such that for every $\epsilon > 0,$ there exists a positive integer $N$ such that if $i,j > N$ , then $|x_{n_i+n_j}-L| < \epsilon.$