measure theory - Using Fatou's Lemma

Let $(X,\Sigma,\mu)$ be a measure space and let $\{E_n\}_{n=1}^\infty$ be a sequence of sets in $\Sigma$. I try to show that $$\lim_{m\to \infty} \mu(\cap_{n=m}^\infty E_n)\leq \liminf \mu(E_n)$$ by using the Fatou's lemma.

Attempt: Let $f_n=\chi_{E_n}$. By Fatou's lemma, we have $\liminf\int f_n \ge\int\liminf f_n$. Clearly, $\liminf\int f_n=\liminf \mu(E_n)$. So it remains to show that $\int \liminf f_n \ge \lim_{m\to \infty}\mu(\cap_{n=m}^\infty E_n)$

I know this link has the same question but I want to conclude my attempt. Thanks!

Answer

Hint: Since $f_n$ only takes the values $0$ and $1$, the same is true of $\liminf f_n$. Hence $\liminf f_n$ is itself the characteristic function of some set. Try to find out which set.