Why is this set club?

Question

I am currently reading a proof on properties of stationary sets and one step of the proof does not make a whole lot of sense to me. The proof asserts that

If $\kappa$ is a regular cardinal and $\alpha < \kappa$, the set $C_{\alpha} = \kappa \smallsetminus \alpha$ is club.

It is pretty clear to me that $C_{\alpha}$ is unbounded, but if $\xi < \kappa$ is some limit ordinal such that $\sup(C_{\alpha} \cap \xi) = \xi$, it is unclear to me why $\xi \in C_{\alpha}$.

If it matters at all, I am reading a proof of the claim "If $\kappa$ is a regular cardinal and $S \subseteq \kappa$ is stationary, then $|S| = \kappa$."

Answer 1

HINT: If $C_\alpha\cap \xi$ is unbounded (or even nonempty) in $\xi$, then some element of $\xi$ must be bigger than $\alpha$. What does that say about $\xi$ and $\alpha$?

That is: think about what "$\kappa\setminus \alpha$" looks like in terms of the ordering on the ordinals.