Suppose that for every $n\in\mathbb{N}$ the $V_n$ is a non-empty closed subset of a compact space $X$ and suppose that $V_{n+1}\subset V_n$,I have to show that $\bigcap_{n=1}^{\infty}V_n\neq \emptyset$
I'm not sure about my solution as follows:
So suppose that the intersection is empty, $\bigcap_{n=1}^{\infty}V_n = \emptyset$.
Now let $U_n=X\setminus V_n$ then the $U_n$ would form a covering of $X$ as $\bigcup U_n=\bigcup(X\setminus C_n)=X\setminus(\cap C_n)=X$
Then we have a finite subcovering $\{U_1,U_2,.....U_n\}$
Am I on the right track with this?
Thanks for any help