Currently reading up on some measure-theoretic probability, and got stuck on the following problem from an earlier exam in a probability theory course.
Problem 3
Let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with finite variance $\text{Var}(X_k) < \infty$. Prove that for any $x\in\mathbb{R}$, the limit $$ \lim_{n\to\infty} P(X_1 + X_2 + \ldots + X_n \leq x) $$ can only take three values. Which 3 values? Find conditions under which the limit takes each of these three values.
My attempt
So far, I've been able to show that $E[X_k] > 0$ implies that the above limit is 0 by using the law of large numbers. In the case where $E[X_k] < 0$, a similar argument show that the limit goes to 1 instead. I have no idea what happens in the case where $E[X_k] = 0$.
How can one show that there are only 3 possibilities, and find conditions for when the last possibility occurs?