First of all a couple of definitions which might be different to the standard ones:
A function is called cofinal if $f:\alpha\mapsto \beta$is such that $\sup\{ f(\gamma):\gamma<\alpha\}=\beta$
Secondly, cf$(\kappa)$ by the $\min\{\alpha:\exists f: \alpha\mapsto \beta\}$ where $f$ is cofinal. An ordinal is called regular if $cf(\alpha)=\alpha$, and singular if it is not regular.
The problem now asks: Prove that there exist singular cardinals of every regular cofinality.
So say we are given $\kappa$ with the property that $\kappa=$cf$(\kappa)$, and then we can consider $\aleph_{\kappa+\kappa}$. Then we first show that there exists a cofinal function that maps $\kappa$ into $\aleph_{\kappa+\kappa}$, just let $f(\alpha)=\aleph_{\kappa+\alpha}$ by the way addition works we see that $\sup\{f(\alpha):\alpha<\kappa\}=\aleph_{\kappa+\kappa}$. However I do not know how to show that this is the least one that works. I am assuming that I must use the regularity of $\kappa$ somehow but I dont know how.
Thanks.