Let $X$ be a compact Hausdorff space and let $\{U_n\}$ be a countable collection of subsets that are open and dense in $X$. Show that the intersection $$\bigcap\limits_{n=1}^\infty U_n$$ is dense.
I tried to show that the closure of this intersection is equal to the intersection of the closures of each set, but I'm not getting anywhere.
Any help would be greatly appreciated.