Say that a set C \subseteq \omega_1 is club if there is a function f: \omega_1 \rightarrow \omega_1 such that C=\{ \beta \in \omega_1 | f[\beta] \subseteq \beta \}. Show that the intersection of countably many club subsets of \omega_1 is nonempty.

Notation: \omega_1 denotes the first uncountable ordinal. We may us the Countable Principle of Choice, but not the Axiom of Choice.