Let $Y$ be a continuous random variable with density function $f_Y(y)=e^{-y}, y > 0$. Consider the sequence $\{X_n\}$, given by $X_n=e^nI_{\{Y>n\}}, n =1,2,\cdots$
How can I show that $\{X_n\} \to 0$ in probability?
My work:
So, I want to show that $P(|X_n-0|>\epsilon) \to 0$ as $n \to \infty$.
$P(|X_n| > \epsilon)=P(X_n > \epsilon)=1-P(X_n \le \epsilon)=1-\int_0^\epsilon\Sigma_1^Y(e^n)dx$
However, I do not think my last equality makes much sense. How would you proceed?