I'm working on the problem below. I've proved one side of it, but I need help on the other side.
Consider $X_1, X_2, \ldots$ as independent random variable where: $\Pr(X_n = k) = (1-p_n)p_n^k$ for $k = 0, 1, 2, \ldots$ ($p_n > 0$)
The goal is to show:
$X_n \rightarrow 0 \text{ almost surely IF and Only IF }\sum_{n} P_n < \infty$
1) part 1: $\sum_{n} P_n < \infty \rightarrow X_n \rightarrow 0 $
To prove this I can simply show that $\sum_{K = 0}^{\infty}P(X_n) = K = 1 < \infty$ then by using Borel Cantelli we know this implies that $P(|X_n| > \epsilon, \text{infinitely often }) = 0 \rightarrow X_n \text{ converges to 0 almost surely}$
Could you guide me whether my solution to first part is correct. Also,could you guide me on the second part?
I appreciate your help.