Set $X$ to be a compact topological space. Let ${V_\alpha}$ be a system of the closed subsets of $X$ where $\bigcap\limits_{\alpha\in ℤ}{V_\alpha}≠\emptyset$ ($\alpha$ is finite). Show $\bigcap\limits_{∞}{V_\alpha}≠\emptyset$.
I'm having a terrible time getting to grips with topology after just starting to study it- hopefully I'll cross the bridge and get it soon! But this question has me truly stumped.