RO-NMO 2015 Problem 113

Let be two nonnegative real numbers $a,b$ with $b>a,$ and a sequence $\left( x_n \right)_{n\ge 1}$ of real numbers such that the sequence $\left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1}$ is bounded.

Show that the sequence $\left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1}$ is convergent.