some basic cardinal arithmetic on $\text{cf}(\aleph_{\omega_1})$ I'm reading The Joy of Sets by K. Devlin, by self-study. I've just seen a statement $\text{cf}(\aleph_{\omega_1})=\omega_1$ without proof, but I think this is slightly harder to prove than more obvious one, $\text{cf}(\aleph_{\omega})=\omega$. 
Concretely, $\text{cf}(\aleph_{\omega_1})\le \omega_1$ is trivial, but.. how about the other direction? I've tried and I guess $\aleph_{\omega_1}^{\aleph_0}=\aleph_{\omega_1} $ (as cardinal exp.) implies the result desired. How can I prove the last identity, or is there a simpler proof on the cofinality?

This is a consequence of a more general fact:
Suppose $\gamma$ is a limit ordinal and I have a sequence of ordinals $\langle \alpha_{\eta}:\eta<\theta\rangle$, with each $\alpha_\eta<\omega_\gamma$. Then this induces a sequence of ordinals $\langle \beta_\eta: \eta<\theta\rangle$, each $<\gamma$, as follows: $\beta_\eta$ is the least $\delta$ such that $\alpha_\eta<\omega_\delta$.
The fact: if $\langle \alpha_\eta: \eta<\theta\rangle$ is cofinal in $\omega_\gamma$, then $\langle \beta_\eta: \eta<\theta\rangle$ is cofinal in $\gamma$.
Setting $\gamma=\omega_1$ addresses your specific question.
