Can we find a sequence $u_n$ of positive real numbers such that $\sum_{n=1}^\infty u_n^2$ is finite, yet $\sum_{n=1}^\infty ({u_1+u_2+...+u_n\over n})^2$ is infinite ?

After several attempts, I think this is not possible, but I can't prove that the finiteness of the first sum implies the finiteness of the second sum.