IT-CESENATICO 2004 Problem 5

Decide if the following statement is true or false: For every sequence $\{x_n\}_{n\in \mathbb{N}}$ of non-negative real numbers, there exist sequences $\{a_n\}_{n\in\mathbb{N}}$ and $\{b_n\}_{n\in\mathbb{N}}$ of non-negative real numbers such that: (a) $x_n = a_n + b_n$ for all $n$ ; (b) $a_1 + \cdots + a_n \le n$ for infinitely many values of $n$ ; (c) $b_1 + \cdots + b_n \le n$ for infinitely many values of $n$ .