# weakly compact cardinals - lemma for stationnary reflexion

Recalling that an uncountable cardinal $$\kappa$$ such that $$\kappa \rightarrow (\kappa)^2_2$$ (using Ramsey theory notation), is an inaccessible cardinal, called weakly compact.

We also know that $$\kappa$$ is weakly compact if and only if it is inaccessible and there are no $$\kappa$$-Aronszajn trees.

I want to prove that

If $$A$$ is a set of infi nite cardinals such that for all regular $$\lambda$$, $$A\cap\lambda$$ is not stationary in $$\lambda$$, then there is a one-to-one regressive function $$f$$ on $$A$$, i.e., for all $$\alpha \in A$$, $$f(\alpha) < \alpha$$.

This would be used later to prove the stationary reflection property of weakly compact cardinals : If $$\kappa$$ is weakly compact and $$S\subseteq\kappa$$ is stationary, then there is a regular $$\lambda<\kappa$$ such that $$S \cap \lambda$$ is stationary in $$\lambda$$.

I'm don't really know how to attack this problem. I tried a constructive approach for $$f$$. We know that for any regular $$\lambda$$, that there is $$C_\lambda$$ a club subset of $$\lambda$$ which do not intersect $$A\cap\lambda$$. I guess this should help me build $$f$$. Any hint is welcome. Thanks