 Supremum convergence for decreasing continuous functions

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 $\{f_n\ge t\}$ 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.