Sum of independent Possion random variables tends to its expectation almost surely? This is Exercise 2.3.3 of Durrett's book "Probability: Theory and examples": 

Let $X_n$ be independent Poission r.v's with $EX_n=\lambda_n$, and let $S_n=X_1+\cdots+X_n.$ Show that if $\sum\lambda_n=\infty,$ then $$\frac{S_n}{\mathbb ES_n}\overset{a.s.}\longrightarrow 1$$ 

By Chebyshev's inequality, it is not hard to see that $S_n/\mathbb ES_n\to 1$ in probability. And our aim is to show $$P\left(|S_n-\mathbb ES_n|>\delta\mathbb ES_n \text{ i.o.}\right)=0$$ for all $\delta>0$. By Chebyshev's inequality, I can find a subsequence of $S_n$ satisfying this property. But I don't know how to extend it to the whole sequence.

You can decompose your random variables $X_i$ into independent Poisson random variables with smaller $\lambda$. So you can construct another sequence which has bounded parameters. You apply your result on the modified sequence. The sum of variances will be the same like the sum of variances on the original sequence, so the divergence of the sum of variances is maintained.
