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? 

Let $X_n$ be $-n$ with probability $p_n$ and $e^{n}$ with probability $1-p_n$ where $p_n=e^{n}/(n+e^{n})$ . To show that this example works note that $\sum (1-p_n)<\infty$ and use Borel Cantelli Lemma. 
