= König bound for the Gimel function
{c}
{title2=$\kappa^{\operatorname{cf}(\kappa)}>\kappa$}
Choose cofinal <ordinals> $\alpha_i<\kappa$, indexed by $\mu=\operatorname{cf}(\kappa)$, and set $a_i=|\alpha_i|$. Then $\sum_i a_i=\kappa$ and $a_i^+\leq\kappa$. The <König theorem for cardinal numbers> gives $\kappa<\prod_i a_i^+\leq\kappa^\mu$. The estimate holds for every infinite <cardinal number>, regular or singular, under the usual <axiom of choice> convention for <cardinal arithmetic>.
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