Show that $S_n$ converges almost surely to $\infty$

Question

Suppose $X_n$ are independent random variables with $P(X_n=-n^2)=\dfrac{1}{n^2}$, $P(X_n=-n^3)=\dfrac{1}{n^3}$ and $P(X_n=2)=1-\dfrac{1}{n^2}-\dfrac{1}{n^3}$ for all $n\geq2$. Let $S_n=\sum_{i=1}^nX_i$. Show that $S_n\to\infty$ almost surely.

I cannot find any result relating sums of random variables to variances or means. Help is appreciated.

Answer 1

Hint: use the Borel-Cantelli lemma to show that $$P(X_n \ne 2 \text{ i.o.}) = 0.$$ (Independence is not needed,)

Answer 2

I will continue @Nate Eldredge's answer and finish this proof.

\begin{equation} \sum\limits_{n\geq1}\mathbb{P}(X_n \neq 2) = \sum\limits_{n\geq1}\mathbb{P}(X_n =-n^2 or \, -n^3) \leq \sum\limits_{n\geq1}\frac{1}{n^2} < \infty \end{equation} Then Borel-Cantelli lemma, we have the desired result. \begin{equation} \mathbb{P}(X_n \neq 2 f.o.) = 1 \end{equation}That is, $\exists N \in \mathbb{N}, \forall n > N, X_n = 2 \; a.s.$ Then for $a.e. \omega\in\Omega$ \begin{equation} S_n = \sum\limits_{i=1}^N X_i+\sum\limits_{i=N+1}^n X_n > \sum\limits_{i=N+1}^n X_n+c =\infty \end{equation}Note the first part in the summation is finite, let's say it is $c\in \mathbb{R}$