Cauchy sequence of random variable

Question

I know (from calculus) the Cauchy convergence theorem for sequence of real numbers...how to show it with random variables?

I have $S_n=\sum_{i=1}^n X_i;$ $X_i$ being iid centered r.v., $S_n$ is a Cauchy sequence in $L^2$, I have to show it converges to $S$ ($E((S_n-S)^2)=0$)

I have to show also that $E(\sum_i^\infty X_i^2)<\infty$ (this should be easy) and that $X_n\rightarrow 0 (a.s.)$ (here I need help)

Answer 1

Because $E(\sum_i^\infty X_i^2)<\infty$, so $\sum_i^\infty X_i^2<+\infty, a.s$. So $X_i^2\rightarrow 0, a.s$ as well as $X_i$.