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 ?

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 $E[Y^2]=E[\liminf Y_n^2]\leq \liminf E[Y_n^2]\leq M$
