Let $A$ be a set of infinite cardinals. Assume that for every regular $\lambda$, the subset $A \cap \lambda$ of $\lambda$ is not stationary. Then I want to prove that there is an injective function $g$ on $A$ which satisfies $g(\alpha) < \alpha$ for all $\alpha \in A$.
This is an exercise in Kunen's set theory. I dont' understand if $g$ should be a map from $A \to A$. But then the claim would be false if $A = \{\kappa\}$. But if $g$ may have arbitrary ordinal values, then the claim is somehow trivial, also without the assumptions. So I don't know what this exercise about.