Question 151
Let $a \geq 0$ and let $\left(x_{n}\right)_{n \geq 1}$ be a sequence of real numbers. Show that if the sequence $\left(\frac{x_{1}+\cdots+x_{n}}{n^{a}}\right)_{n \geq 1}$ is bounded, then the sequence $\left(y_{n}\right)_{n \geq 1}$, defined by $y_{n}=\frac{x_{1}}{1^{b}}+\frac{x_{2}}{2^{b}}+\cdots+\frac{x_{n}}{n^{b}}$, is convergent for every $b\gt a$.