Assume that $X_1,X_2,\ldots$ are independent random variables (not necessarily of the same distribution). Assume that that $Var[X_n]>0$ for all $n$. Assume also that
$\sum_{n=0}^\infty \frac{Var[X_n]}{n^2}<\infty,$
that
$\frac{1}{n}\sum_{i=1}^n(X_i-E[X_i])\to 0 \textrm{ almost surely as $n\to\infty$},$
that
$E[X_n]>0 \textrm{ for all $n$},$
and that
$\liminf_{n\to\infty} E[X_n] > 0.$
How can we prove that $\sum_{i=1}^n X_i \to \infty\text{ almost surely as $n\to\infty$?}$ This seems to intuitively make sense, but a formal proof escapes me. Also, what can we say if $E[X_n] = 0$ for all $n$ and $\lim_{n\to\infty} E[X_n] = 0$?