## club set

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.