Define Lim($\omega_1$)={$\delta<\omega_1\ :\ \delta$ is a limit ordinal}. Assume that $\left<A_\alpha:\alpha\in\text{Lim}(\omega_1)\right>$ is a sequence satisfying the following:
(1) $\forall\alpha\in\text{Lim}(\omega_1)[A_\alpha\subseteq\alpha\land\sup{(A_\alpha)}=\alpha]$;
(2) for every uncountable $X\subseteq\omega_1$, there exists $\alpha\in\text{Lim}(\omega_1)$ such that $A_\alpha\subseteq X$.
Prove that for every uncountable $X\subseteq\omega_1$, $\{\alpha\in\text{Lim}(\omega_1):A_\alpha\subseteq X\}$ is stationary.
I was just reading about set theory and came across this exercise. I know that $S\subseteq\delta$ is called stationary in $\delta$ if $\forall C\in\text{club}(\delta)$ (or called Cub($\delta$)) we have $S\cap C \neq \phi$. And $\text{club}(\delta)$ are the subsets of $\delta$ who have a club as a subset. How do we go on to prove that these sets are stationary? Any direction or hint would be appreciated. Thank you.