Product of Indicator function sequence and integrable random variable has expectation zero in convergence.

Question

Let $X$ be a positive integrable random variable i,e, $E[X]< \infty $. Define $X_n = X\mathbb{I}_{X>n}$. Show that $\lim_{n\rightarrow \infty} E[X_n] = 0$.

I tried the following: $E[X]< \infty $, implies $P[X<\infty]=1$ which implies $\lim_{n\rightarrow \infty}P[X>n]=0$. I know that somehow I have to use Cauchy-Schwarz or Holders inequality. But not sure how to proceed from here. Please help.

Answer 1

Hint: Write $X_n = X - X \mathbb{I}_{X\le n}$ and appeal to the monotone/dominated convergence theorem.