Strong law of large numbers with $\sum_{n=0}^\infty \frac{Var[S_n]}{n^2}<\infty$

Question

Given independent real random variables $X_1,X_2,... \in L^2$ with $$\sum_{n=0}^\infty \frac{Var[S_n]}{n^2}<\infty$$ (here $S_n := X_1+...+X_n$).

How do you show that $(X_n)_{n \in \Bbb N}$ fulfills the strong law of large numbers?

Answer 1

See Chapter X of Feller, "Introduction to Probability Theory and its Applications, Vol. I, Wiley (1968). He proves the theorem on pages 259-261.