I'm trying to solve this exercise. Can anybody please help me:

If $\kappa$ ist an infinite cardinal number with $cf(\kappa) = \kappa$ and for all $\mu < \kappa$ the inequality $2^{\mu} \leq \kappa$ holds, then I should prove

$\sum_{\mu < \kappa,\ \mu \in Card} \kappa^\mu = \kappa$

I've already found out that $\sum_{\mu < \kappa,\ \mu \in Card} \kappa^\mu = \sup_{\mu < \kappa,\ \mu \in Card} \kappa^\mu$, so it is sufficient to prove that this supremum is equal to $\kappa$. But I can't see why this is true.



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