Almost sure convergences of series of Poisson random variables Let $X_i$ be independent Poisson with mean $\lambda_i$. Show that 
i)$\sum_1^\infty\lambda_i<\infty$ implies $\sum_1^\infty X_i$ converges almost surely to a finite limit.
ii)$\sum_1^\infty\lambda_i=\infty$ implies $\sum_1^\infty X_i=\infty$ almost surely
I could prove (i) by using Kolmogorov one series lemma just setting $Y_i=X_i-\lambda_i$, but I could not prove the second part. Any help/hint/solution for (ii) is appreciated. Thanks in advance

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*$$\infty > \sum_{n} \lambda_n = \sum_{n} E[X_n] = E[\sum_{n} X_n]$$
$$\to \infty > \sum_{n} X_n$$


*$$E[X_n] = Var[X_n] = \lambda_n$$


Use contrapositive of Kolmogorov's three-series theorem to show that $\forall A > 0$, one of the three series diverges.
