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


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