If $X\geq0$ is a random variable, show that $\lim\limits_{n\to\infty}\frac1nE\left(\frac{1}{X}I\left\{X>\frac{1}{n}\right\}\right)=0$

Question

If $X\geq0$ is a random variable then show that:$$\lim_{n\to\infty} \frac{1}{n} \cdot E\bigg(\dfrac{1}{X}I\bigg\{X>\dfrac{1}{n}\bigg\}\bigg)=0$$

A hint would be most appreciated. I have studied measure theory, but I presume this can be solved using simple analysis.

Answer 1

Let $X_n=\frac{1}{n}\frac{1}{X}\mathbf{1}_{[X>\frac{1}{n}]}$. Show that $|X_n|<1$ and $X_n$ converges to $\mathbf{0}$. Then use Dominated Convergence Theorem..