Prove that $\bigcap_{k = 1}^\infty C_k$ is also compact and connected. [duplicate]

if your intersection (denote $C$) is not connected, you can represent is as a unit of two disjoint both relatively open and relatively closed nonempty subsets of $C$, say $A$ and $B$.Then you can find two open disjoint subsets $U,V$ in $X$ such that $U\cap C = A$, $V\cap C = B$ (Well, since it is a Hausdorff space and $A$ and $B$ are closed in a compact subset, hence compact, you always can find two open sets $U',V'$ such that $A\subset U', B\subset V'$ and $U'\cap V' = \emptyset$. Also by the definition of a subspace topology, you can find open in $X$ sets $U'',V''$ such that $U''\cap C = A, V''\cap C = B$ (they needn't be disjoint, though), then take $U = U'\cap U'', V = V'\cap V''$). Then $U\cup V$ is an open set containing $C$. Hence there exists $n\in \mathbb{N}$ such that $C_n\subset U\cup V$, but this means that $C_n$ is not connected (a contradiction)! Seems like that... Are there any mistakes? :\