I am trying to prove that if $cof(\kappa)=\omega$, then the intersection of two clubs doesn't have to be a club.
My idea is to make two sequences which would have empty intersection.
So by the definition of cofinality,
I can take sequence $(\alpha_n)_{n \in \omega}$ which is unbounded in $\kappa$ and then I was thinking of taking two subsequences, say $(\alpha_{2n})_{n \in \omega}$ and $(\alpha_{2n+1})_{n \in \omega}$ and if for $C$ and $D$ I take the set consisting of all $\alpha_{2n}$, resp. set of all $\alpha_{2n+1}$, again unboundedness is pretty much clear, but how can I prove closedness?
Intiutively, the reason $C$ and $D$ are closed is that they are "discrete" - their only possible limit point would be $\kappa$ itself. (This goes back to the picture I gave in this answer to your previous question.)
This is informal, of course, but the idea is correct. Suppose $\lambda<\kappa$ is a limit cardinal; can $\lambda$ be a limit point of $C$? That is, can $C\cap\lambda$ possibly be unbounded in $\lambda$? (HINT: how big is $C\cap\lambda$?) What does this say about the closedness of $C$?