During my research, I was required to prove a particular result. I shall just ask what I needed for my result to hold.
Let $X_n$ be a sequence of random variables that are integrable and suppose we have for some constant $C>0$ that $$\lim_{n \to \infty} P[X_n \leq C] = 0$$
Then does it follow that $$\lim_{n \to \infty}E[X_n1_{\{X_n \leq C\}}] = 0$$ ?
I don't know if the result true or not. However as a start, I was able to show that $$\limsup_{n \to \infty}E[X_n1_{\{X_n \leq C\}}] \leq 0 $$
However I have no idea how to proceed from here and show $$\liminf_{n \to \infty}E[X_n1_{\{X_n \leq C\}}] \geq 0 $$
This wouldn't be a problem if $X_n$ were non negative but since that is not given, I am stuck here. I appreciate hints or counterexamples to this. My guess, if I had to make one, is that this result isn't true...
Consider $X_n$ such that $P(X_n= n^2)= 1-\frac{1}{n}$ and $P(X_n=-n^2)= \frac{1}{n}$.
This has $\displaystyle\lim_{n \to \infty} P[X_n \leq C] = 0$ for all $C \gt 0$ but $\displaystyle\lim_{n \to \infty}E[X_n1_{\{X_n \leq C\}}] = \lim_{n \to \infty}-n=-\infty$.