SCHWEITZER 1970 Problem 11

Let $\xi_1,\xi_2,...$ be independent random variables such that $E\xi_n=m>0$ and $\textrm{Var}(\xi_n)=\sigma^2 < \infty \;(n=1,2,...)\ .$ Let $\{a_n \}$ be a sequence of positive numbers such that $a_n\rightarrow 0$ and $\sum_{n=1}^{\infty} a_n= \infty$ . Prove that $$P \left( \lim_{n\rightarrow \infty} \sum_{k=1}^n a_k \xi_k =\infty \right)=1.$$

P. Revesz