Can there exist $\{X_n\}_{n\ge 1}$ such that $X_n \to -\infty$ a.s. and $EX_n\to 0$? Can there exist variables $\{X_n\}_{n\ge 1}$ such that  $X_n \to -\infty$ a.s. and   $EX_n\to 0$?
How can we prove or disprove this analytically? 

Here is a slightly simpler example than Kavi's, which avoids Borel-Cantelli.
Fix a random variable $U \sim U(0,1)$ and let
$$X_n = \begin{cases} n^2-n ,& U < \frac{1}{n} \\ -n , & U \ge \frac{1}{n}. \end{cases}$$
You can easily check that $E[X_n]=0$.  And on the event $U > 0$, which has probability 1, we have $X_n = -n$ for all sufficiently large $n$ and therefore $X_n \to -\infty$.  
