A problem on random series Suppose $X_1, X_2,\dots, $ be an independent sequence of random variables and $E[X_n] = 0 \forall n$ and $\sum_{n=1}^{\infty} \operatorname{Var}(X_n) < \infty$. I need to prove that $\operatorname{Var}\left(\sum_{n=1}^{\infty} X_n \right) < \infty$. How to proceed ?

The series $\sum_{n=1}^\infty X_n$ converges in mean square. Observe that
$$
\operatorname E\biggl|\sum_{n=1}^NX_n-\sum_{n=1}^MX_n\biggr|^2=\sum_{n=M+1}^N\operatorname EX_n^2<\varepsilon
$$
for large values of $N$ and $M$ since $\sum_{n=1}^\infty\operatorname EX_n^2<\infty$.
If the series of random variables converges in mean square, then the sequence of variances converges and
$$
\lim_{n\to\infty}\operatorname{Var}\biggl[\sum_{k=1}^nX_k\biggr]=\operatorname{Var}\biggl[\sum_{k=1}^\infty X_k\biggr]
$$
(see this question for a proof).
