Showing that $ |\operatorname{club}_\kappa| > \kappa $

Question

I'm trying to solve the following exercise:

Show that $ |\operatorname{club}_\kappa| > \kappa $

Worded differently:

Show that there are more than $ \kappa $ closed and unbounded subsets of $\kappa$

I think this might be some sort of standard diagonal argument, however, I have no idea where to begin

I would appreciate some help

Answer 1

Here is a simple construction. Let $L$ be the set of all limit points of $\kappa$. It’s a club of size $\kappa$. For each $X\subseteq L$, take $$C=\{\alpha+1\mid\alpha \in X\}\cup L\;;$$ this is again a club.