= Supremum convergence for decreasing continuous functions
{title2=$\sup_K f_n\downarrow\sup_K\inf_n f_n$}
On a nonempty <compact Hausdorff space>, let continuous nonnegative functions decrease pointwise. Their suprema decrease to a finite number $L$. For every $t<L$, the sets $\{f_n\geq t\}$ are nonempty nested closed sets; compactness gives a common point with limit value at least $t$. Thus the supremum of the pointwise limit is $L$. The limit need not be continuous, and this conclusion does not claim uniform convergence as in the <Dini theorem>.
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