Prove that exist $n_0: F_{n_0}=\emptyset$ in a compact space

Question

Let $X$ be a compact space and $F_n$ is close, $F_{n+1} \subset F_n$ and $\displaystyle \bigcap_{n=1}^\infty F_n = \emptyset$.

Question 1: Prove that exist $n_0: F_{n_0}=\emptyset$.

Question 2: State the similar problem above for the sequence of open sets

I have tried to prove question 1, I have no idea to use the information $X$ is a compact space, so I didn't find the solution

For question 2, can I apply question 1 with $X \setminus F_n$ open. Do you have any idea for it ? Thank you.

Answer 1

Hint

Let $U_n=X\setminus F_n$. Then

$$\bigcup_{n=1}^\infty U_n=X\setminus\bigcap_{n=1}^\infty F_n=X\setminus\emptyset=X$$