In our probability theory class, we are supposed to solve the following problem:
Let $X_n$, $n \geq 1 $ be a sequence of independent random variables such that $ \mathbb{E}[X_n] = 0, \mathbb{Var}(X_n) = \sigma_n^2 < + \infty $ and $ | X_n | \leq K, $ for some constant $ 0 \leq K < + \infty, \ \forall n \geq 1$.
Use martingale methods to show that $ \sum \limits_{n = 1}^{\infty} \ X_n \ \mbox{ converges } \mathbb P-\mbox{a.s.} \ \Longrightarrow \ \sum\limits_{n = 1}^{\infty} \ \sigma_n^2 < + \infty .$
Could anybody give me a hint? Thanks a lot for your help!
Regards, Si