I am trying to solve a problem from the book(A Course in Probability Theory" by Chung, Third Edition page 132). The problem says:
Let $\{X_n\}$ be a sequence of independent random variables with $E(X_n)=0$. Suppose $\forall n: Var(X_n)=\sigma_n^2<\infty$ and $s^2_n=Var(\sum_{j=1}^{n}X_j)=\sum_{j=1}^{n}\sigma_j^2 \rightarrow \infty.$ If $a_n=s_n(log s_n)^{0.5+\epsilon}, \epsilon>0,$ then by Dini's theorem we have $$\sum_{n=1}^{\infty}\frac{E(X_n^2)}{a_n^2}= \sum_{n=1}^{\infty}\frac{\sigma_n^2}{s_n^2(log s_n)^{1+2\epsilon}}<\infty.$$ How can apply Dini's theorem? Thank you for help me.