Kolmogorov's convergence theorem: summable variances force a.s. convergence

Let $\{X_n\}$ be a sequence of independent random variables such that $E[X_n]=0$ and $\operatorname{Var}(X_n)=\dfrac1{n^2}$, $n=1,2,\dots$. Let $E=\{\omega\mid\sum_nX_n(\omega)\text{ converges}\}$. Then which of the following is always correct?