Let $\mu < \kappa$ be infinite cardinals. A set $C$ is called a $\mu$-club in $\kappa$, if it is unbounded in $\kappa$ and contains all its limit points of cofinality $\mu$.
Now let $T \subset S := \{ \alpha < \kappa \mid cf(\alpha) = \mu \}$ be stationary (in the usual sense). I'd like to argue, that $T$ an $C$ have an element in common.
The idea is to recursively construct a sequence $(C_\alpha \colon \alpha \in On)$ as follows:
- $C_0 = C$
- $C_{\alpha+1} = C_\alpha \cup C_\alpha'$
- $C_\lambda = \bigcup_{\alpha < \lambda} C_\alpha$
where $C_\alpha'$ is the set of limit points of $C_\alpha$.
Let $C^*$ be the least $C_\alpha$ such that $C_\alpha = C_{\alpha+1}$. Now $C^*$ is a club and thus $C^* \cap T \neq \emptyset$. To finish the proof, note that $C^* \cap S = C \cap S$ (as seen by an induction on $C_\alpha$).
Can someone please check this proof or refer me to another proof of this result?