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

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