= Cofinality of an increasing ordinal supremum
{title2=$\operatorname{cf}(\sup_{\xi<\alpha}\gamma_\xi)=\operatorname{cf}(\alpha)$}
If $\alpha$ is a nonzero <limit ordinal> and $(\gamma_\xi)_{\xi<\alpha}$ is strictly increasing, then $\operatorname{cf}(\sup_{\xi<\alpha}\gamma_\xi)=\operatorname{cf}(\alpha)$. A cofinal subsequence of the index order gives one inequality. Conversely, a cofinal family in the supremum determines a cofinal family of indices by choosing an index past each of its values. This also explains the <cofinality> of limit-indexed <aleph numbers>.
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