Let $(a_n)_{n \ge 1}$ be a sequence of real numbers satisfying the properties: the sequence $x_n=\sum\limits_{k=1}^n a_k^2$ is convergent; the sequence $y_n=\sum\limits_{k=1}^n a_k$ is unbounded. Prove that the sequence $(b_n)_{n \ge 1}$ given by $b_n=\{y_n\}$ is divergent. Note: $\{ x \}$ denotes the fractional part of $x.$
Problem
Source: Romanian National Olympiad 1998 - Grade 11 - Problem 2
Tags: real analysis, Sequences, divergence