Let $\{X_i\}$ be a uniformly bounded sequence of independent random variables. Does $\sum_{i=1}^{\infty}X_i-E[X_i]$ converges or diverges, depending on whether $\sum_{i=1}^{\infty}\sigma_i^{2}$ converges or diverges?

I have looked everywhere for a proof with little success. Can someone provide a reference or a proof? Thank you.