I'm reading Jech's Set Theory book and I encountered this problem (Chapter 3 exercise 12).
If $\aleph_\alpha$ is an uncountable limit cardinal, then $\text{cf } \omega_\alpha = \text{cf } \alpha$; $\omega_\alpha$ is the limit of a cofinal sequence $\langle \omega_{\eta} : \eta < \text{cf }\alpha \rangle.$
My question is, why is the last assertion true? From what I understand, the limit should be $\omega_{\text{cf } \alpha}$ (by definition of $\omega_{\alpha}$ when $\alpha$ is a limit ordinal)