Suppose $A$ is a non stationary set of $\omega_1$. Define by induction the following sequence of sets:\
$A_{\alpha+1} = A_{\alpha}'$ [$X'$ is the subset of $X$, of all points the are limits of sequences from $X$]
For limit stage we take the intersection.
Is it true that for some $\alpha < \omega_1$, $A_{\alpha}$ is null ?
No. Take a fast growing continuous $h:\omega_1 \to \omega_1$ such that $\operatorname{otp}(h(\alpha+1)-h(\alpha)) \geq \omega^\alpha$ for each $\alpha$. Consider the non-stationary set $X = \omega_1 - \{h(\alpha) : \alpha \lt \omega_1\}$. Since it takes $\alpha$ derivatives to exhaust $\omega^\alpha$, the interval $[h(\alpha)+1,h(\alpha+1)) \subseteq X$ cannot be exhausted in fewer than $\alpha$ steps.
