I'm trying to solve the following exercise:
Suppose $X_1$ and $X_2$ are i.i.d. random variables with common $\mathcal{N}(0,1)$ normal distribution. Define $Y_n = X_1(\frac{1}{n} + |X_2|)^{-1}$. Use Fubini's theorem to verify that $\mathbb{E}(Y_n) = 0$. Note that as $n\to \infty$, $Y_n \to Y = X_1 |X_2|^{-1}$ and that the expectation of $Y$ does not exists, so this is one case where random variables converge but means do not.
I really don't know how to start solving this. Can you give me any hint (not the solution, but hints)? Thank you very much in advance
P.S.: This exercise appears is A Probability Path, by Resnick, chapter 5 (which is about integration and expectation).
P.P.S.: I had an idea, but I do not use Fubini's theorem: given that $X_1$ is independent of $(\frac{1}{n} + |X_2|)^{-1}$ (because the composition is measurable), and then $$ \mathbb{E}(Y_n) = \mathbb{E}(X_1) \mathbb{E} \left[ \left(\frac{1}{n} + |X_2| \right)^{-1} \right] = 0$$