Let $\kappa$ be a regular uncountable cardinal, and let $\lambda < \kappa$ be a regular cardinal. Then the set $E_{\lambda}^{\kappa} := \{ \alpha < \kappa \mid \operatorname{cf}\alpha = \lambda \}$ is stationary in $\kappa$. It is a result that every stationary subset of $E_{\lambda}^{\kappa}$ is the disjoint union of $\kappa$ stationary sets. Now define $S = \{\alpha < \kappa \mid \operatorname{cf}\alpha<\alpha\}.$
In Chapter 8, Jech makes the following claim: "Every stationary subset $W \subseteq S$ is the disjoint union of $\kappa$ stationary sets: By Fodor’s Theorem, there exists some $λ < κ$ such that $W ∩ E_{\lambda}^{\kappa}$ is stationary."
I'm a bit confused by this. Here is what I managed to extract from this problem: Is it true that S is always stationary? In other words, is it always possible for it to contain stationary subsets $W$?
If so, then consider the function $\operatorname{cf}:W \to \kappa$ which gives the cofinality of $\alpha \in S$. By definition this is regressive (i.e. $\operatorname{cf}\alpha < \alpha$) so by Fodor's Theorem, there exists a $\lambda < \kappa$ and a stationary set $T \subseteq W$ such that $\operatorname{cf}\alpha = \lambda$ on $T$. If $\lambda$ is a regular cardinal then I can see that $T \subseteq E_{\lambda}^{\kappa} \cap W$ and so $W\cap E_{\lambda}^{\kappa}$ is stationary as a superset, and we're done. But what if $\lambda$ is not a regular cardinal, and then $E_{\lambda}^{\kappa}$ doesn't make sense?
Any help with this would be appreciated.