Similar to Fodor lemma

Question

Let $\lambda>\aleph_0$ be a regular cardinal such that $S \subseteq \lambda$ is not a stationary subset. Prove that there exists a regressive function $f:S \to \lambda$ such that $|f^{-1}(\alpha)|<\lambda$ for every $\alpha<\lambda$.

Answer 1

HINT: If $C\subseteq\lambda$ is closed, and $\alpha\in\lambda\setminus C$, the set $C\cap\alpha=\{\xi\in C:\xi<\alpha\}$ has a maximum element.