I am following a Set Theory course and I have a doubt on an equality. Following Jech Set Theory in chapter 5 we have:
Which makes sense. In the course I am given:
It would seem thought that the statement stated in Jech hold even when $\kappa$ is not regular since the only fact used is that $\lambda<cf(\kappa)$ (which of course follow from $\lambda<\kappa$ and $\kappa$ being regular). The issue on the other hand how this was presented in my course is the requirement of multiplying each term in the sum by $\xi^+$ which doesn't seem necessary. The reason that was given is that for each cardinal $\xi$ there are $\xi^+$ ordinals that are equipotent to $\xi$. The reason why this doesn't seem necessary is that one doesn't need to take the union of all $\alpha^\lambda$ for all ordinals $\alpha\in \kappa$ but rather just of any cofinal set of ordinals. This is because if a function $f$ has co domain $\alpha$ it also has codomain in $\beta>\alpha$. Am I correct or am I missing something? Can the statement in Jech be generalized to the case where $\kappa$ is not necessarily regular but $\lambda<cf(\kappa)$ without having to add the $\xi^+$ to the sum?