Let $X$ and $Y$ be independent random variables. Suppose $P(X=0)<1$ and $\lim_{z \rightarrow 0}f_{|Y|}(z) > 0$. Prove that $E |\frac{X}{Y}| = \infty$.
I defined a random variable $Z = \frac{X}{Y}$. Then tried to show:
$$\int_{-\infty}^{\infty}zf_{|Z|}(z) dz = \int_{-\infty}^{\infty}z\frac{d}{dz}P(|Z| < z) dz = \int_{-\infty}^{\infty}z\frac{d}{dz}P(|X| < z|Y|) dz = \int_{-\infty}^{\infty}z\frac{d}{dz}\int_{0}^{\infty} \int_{0}^{zy} dx dy$$
But I got stuck here. I'm also not convinced that this is the best way to approach the problem. Any help would be really appreciated
$$E(|\frac X Y|)=E(|X|)E(|\frac 1 Y|)$$ by independence. The first term is positive because $P(|X|>0)>0$ and the second term is $\infty$ because $\lim_{t\to 0}f_{|Y|}(t)>0$.