How to prove $cf(\aleph_{\omega_1})=\omega_1$?

Question

I am reading Jech's & Hrbacek's book "Introduction to set theory" and this is question 2.2 from chaper 9:

Prove $cf(\aleph_{\omega_1})=\omega_1$

Isn't it clear from the fact that $\aleph_{\omega_1} = \lim\limits_{\alpha \rightarrow \omega_1}\aleph_\alpha$?

I am not sure what is there to prove here..

Thanks for any help..

Answer 1

It's true that it is immediately clear that $\operatorname{cf}(\aleph_{\omega_1})\leq\omega_1$, but you still have to argue that it is not strictly smaller.

More generally, however, you can show that if $\delta$ is a limit ordinal then $\operatorname{cf}(\aleph_\delta)=\operatorname{cf}(\delta)$.