= Long chain under eventual domination
{title2=$\langle f_\zeta:\zeta<\kappa^+\rangle$}
For every infinite <regular cardinal> $\kappa$, there is a strictly eventually increasing sequence of $\kappa^+$ <functions> $\kappa\to\kappa$. At stage $0<\zeta<\kappa^+$ enumerate predecessors by $e:\kappa\to\zeta$ and put $f_\zeta(\delta)=\sup_{\eta<\delta}(f_{e(\eta)}(\delta)+1)$. Regularity bounds each value below $\kappa$, while any predecessor's index is eventually included in the supremum.
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