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 ?
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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).
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If you call $Y_n:=\sum_{k=1}^n X_k$ and $Y:=\sum_{k=1}^{+\infty} X_k$, $Y_n \rightarrow Y$ pointwise. Furthermore, by hypothesis $ \exists M>0$ s.t. $\forall n$ $E[Y_n^2]<M$.
Now we use Fatou's Lemma and we obtain $E[Y^2]=E[\liminf Y_n^2]\leq \liminf E[Y_n^2]\leq M$
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