Eventual domination on a regular cardinal
= Eventual domination on a regular cardinal
{title2=$f<^*g$}
For $f,g:\kappa\to\kappa$, write $f<^*g$ when there is $\gamma<\kappa$ such that $f(\delta)<g(\delta)$ whenever $\gamma<\delta<\kappa$. For an infinite <regular cardinal>, suprema of fewer than $\kappa$ <ordinals> below $\kappa$ remain below it, permitting the <long chain under eventual domination> construction.