I am dealing with an Orstein-Uhlenbeck process $X_t$ with its stochastic differential equation being

$$dX_t=(\mu-X_t)dt+\sigma dW_t.$$

I want to show

$$\mathbb{E}\left[\frac{|X_\infty|}{\int_{0}^{\infty}f(X_s)ds}\right]=0,$$

where $f(x)=\mathbb{1}\{x\leq a\}$ for some $a>\mu$. Is there a simple way to prove this? Thanks!