UA-UMO 2023 Problem 104

Let $(x_n)$ be an infinite sequence of real numbers from interval $(0, 1)$ . An infinite sequence $(a_n)$ of positive integers is defined as follows: $a_1 = 1$ , and for $i \ge 1$ , $a_{i+1}$ is equal to the smallest positive integer $m$ , for which $[x_1 + x_2 + \ldots + x_m] = a_i$ . Show that for any indexes $i, j$ holds $a_{i+j} \ge a_i + a_j$ .

Proposed by Nazar Serdyuk