In Jech's Set Theory we have the following beginning of a proof of Solovay's theorem (theorem 8.10):
Let $A$ be a stationary subset of $\kappa$ [regular uncountable]. [By previous analysis] we may assume that the set $W$ of all $\alpha\in A$ such that $\alpha$ is a regular cardinal and $A\cap\alpha$ is not stationary, is stationary. There exists for each $\alpha\in W$ a continuous increasing sequence $\langle a_\xi^\alpha\mid\xi<\alpha\rangle$ such that $a_\xi^\alpha\notin W$ for all $\alpha$ and $\xi$ and $\alpha=\lim_{\xi\to\alpha}a_\xi^\alpha$.
The proof then continues to show there is a $\xi$ such that for any $\eta<\kappa$ the set $\{\alpha\in W\mid a^\alpha_\xi\geq\eta\}$ is stationary, by achieving a contradiction where for certain $\gamma,\alpha\in W$ we have $\gamma<\alpha$ and $a^\alpha_\gamma=\gamma$, which contradicts the last sentence of the quote above.
I understand the contradiction, but I don't see why we can be sure of the existence of $\langle a_\xi^\alpha\mid\xi<\alpha\rangle$ such that $W\cap\{a^\alpha_\xi\mid\xi<\alpha\}=\varnothing$. So why does this sequence exist?
By definition, $A\cap\alpha$ is nonstationary for $\alpha\in W $, so there is a club subset $C_\alpha$ of $\alpha $ disjoint from $A $. By definition, $W\subseteq A $, so the same club $C_\alpha $ works. The $a^\alpha_\xi $ are the increasing enumeration of $C_\alpha $.