Let $\kappa$ be a singular cardinal with $\operatorname{cf}\kappa = \lambda > \omega$.
Let $C = \{ \alpha_{\zeta} \mid \zeta < \lambda \}$ be a strictly increasing continuous sequence of cardinals with limit $\kappa$. I want to show this is a closed unbounded set in $\kappa$.
Obviously it is unbounded, because $\sup C = \kappa$, but how do I show it is closed? Is it to do with continuity? Also I am used to dealing with clubs in regular cardinals, are there any differences for clubs in singular cardinals?
Thanks very much.