We call a subset $X \subseteq \kappa$ of a regular cardinal $\kappa$ ineffable, iff for every family $(A_\alpha \mid \alpha \in X)$ of subsets $A_\alpha \subseteq \alpha$, there is a stationary set $S \subseteq X$, such that $A_\alpha = A_\beta \cap \alpha$ for all $\alpha, \beta \in S$ with $\alpha \le \beta$. Now, $\{\alpha < \kappa \mid cf(\alpha) = \lambda\}$ is claimed not to be ineffable, where $\lambda < \kappa$ is a regular cardinal.
Can anyone give me a hint how to see this?
HINT: For each $\alpha\in\{\alpha<\kappa\mid\operatorname{cf}(\alpha)=\lambda\}$ pick $A_\alpha$ to be a cofinal sequence of order type $\lambda$. If $\alpha<\beta$, can $A_\beta\cap\alpha$ be equal to $A_\alpha$?