Successor of a regular cardinal does not reflect stationary sets

Question

Let $\kappa$ be a regular cardinal. Then supposedly $\kappa^+$ does not reflect stationary sets. That is to say, there exists a stationary set $S\subseteq\kappa^+$ such that for every $\alpha<\kappa^+$ of uncountable cofinality, $S\cap\alpha$ is not stationary in $\alpha$. I'm not sure why this is true. My guess would be to let $$S=\{\alpha<\kappa^+ : \operatorname{cf}(\alpha)=\kappa\}$$ but I'm not sure if this stationary set works or how to show it does work.

Answer 1

Edited because my previous example did not suffice:

Let $\alpha<\kappa^+$ have uncountable cofinality. I'll build a club $C$ of order type $\operatorname{cf}(\alpha)$ consisting only of members of cofinality less than $\operatorname{cf}(\alpha)$. If $A=\{a_{\alpha}\}$ is any $\operatorname{cf}(\alpha)$-sequence with supremum $\operatorname{cf}(\alpha)$, then I will let $C$ be the club in $\alpha$ which is the sequence $\{a_{\alpha}\}$, but adding $1$ to every successor ordinal in $A$. Then your set $S$ does not reflect at $\alpha$. This should suffice.