For $\kappa \geq \omega$, which is a cardinal, then how to show $\kappa^{cf(\kappa)}>\kappa$?
My idea: When $\kappa=\aleph_\omega$, then $cf(\kappa)=\omega$, is $\kappa^\omega>\kappa$? It seems $\kappa^{cf(\kappa)}>\kappa$ is wrong.
Could you help me?
This is known as Koenig's theorem.
In this thread there are several answers about the proof that if $A_i<B_i$ for all $i\in I$ then $|\bigcup A_i|<|\prod B_i|$.
Apply this result to the case where $I=\operatorname{cf}(\kappa)$, $\langle A_i\mid i<\kappa\rangle$ is a partition of $\kappa$ into sets of cardinality $<\kappa$, and $B_i=\kappa$ for all $i$. We have, if so:
$$\kappa=\left|\bigcup_{i\in I}A_i\right|<\left|\prod_{i\in I}B_i\right|=\kappa^{\operatorname{cf}(\kappa)}$$