Let $\kappa>\omega$ be a regular cardinal. Let $C\subseteq\kappa$ be a club. Prove that there exists a function $f:\kappa\rightarrow\kappa$ such that $C_f=\{0<\alpha<\kappa:\forall\xi<\alpha[f(\xi)<\alpha]\}\subseteq C$.
There is a related exercise in Jech which proves for $\kappa>\omega$ regular, and any $f:\kappa\rightarrow\kappa$, that $C_f$ is a club. But the exercise in quote asks for the existence of $f$. I was thinking about defining such an $f$ explictly, but that led to nowhere. Any hint would be appreciated. Thank you.