I have been trying to prove this for the last three days. It's for my Time Series Econometrics homework. I think you'll notice I'm not good at math as I can't even express my solution very well, and I'm sorry for that.
At first I thought that if: (1) every element is finite; (2) there is one element that is greater or equal to any other in the sequence; (3) no element repeats itself for more than a finite number of times, then we would have $\sum \limits_{i=0}^{\infty}{|x_i|}<\infty$, because the terms would converge to zero(sooner or later, hehe).
Believing that $x_i^2$ is a monotonic transformation over $|x_i|$, all the properties that guaranteed the convergence of a series would remain unaltered and thus the proof would be given that $\sum \limits_{i=0}^{\infty}{x_i^2}<\infty$.
But now I've just seen that the convergence of terms to zero does not guarantee the convergence of a series, and got pretty confused about the correct approach to the problem.