Question 188
Let $(x_n)$ be a sequence of real numbers from the interval $(0; 1)$. The sequence of positive integers $(a_n)$ is defined as follows: $a_1 = 1$, $a_{i+1} = m$, where $m$ is the smallest positive integer for which $[x_1 + x_2 + \cdots + x_m] = a_i$. Prove that for any indexes $i, j$ the inequality $a_{i+j} \ge a_i + a_j$ holds.
(Nazar Serdyuk)
(Nazar Serdyuk)