As in the titel we want to proof that $\sum_nX_n<+\infty$ a.s. where $(X_n)_{n\geq 1}$ a sequence of independent r.v.'s all on the same probability space $(E,\mathcal{E},P)$. We aslo know $P(X_n=\frac{1}{n})=P(X_n=-\frac{1}{n})=\frac{1}{2}$. It is also noted that $E[|Z|]\leq E[Z^2]^\frac{1}{2}$ might be usefull.
I guess we have to use Borell Cantelli here but I don't see how?