Problem about stationary set

Question

I have a problem about stationary sets. I want to show $A=\{\alpha < \omega_1 : \alpha$ is limit and $cof(\alpha)=\omega \}$ is stationary in $\omega_1$. I want to show that if $C$ is a club then $C \cap A \neq \emptyset$. how I should proceed? Thanks for you help!

Answer 1

Here's a hint: let $\langle \alpha_n \mid n < \omega \rangle$ be a strictly increasing chain of ordinals in $C$ and consider $\alpha = \sup_{n<\omega}\alpha_n$.