 Solution

Assume the compact space is nonempty. Write $M_n={\sup }_K{f}_n$. Each $M_n$ is finite and attained, and the sequence decreases to a number $L\ge 0$. Since $f\le f_n$, certainly ${\sup }_{K}f\le L$.

For any $t<L$, the sets $E_n=\{x\in K:f_n(x)\ge t\}$ are nonempty closed sets, and $E_{n+1}\subseteq E_n$. The finite intersection property and compactness give a point in their intersection. At that point $f\ge t$, so ${\sup }_{K}f\ge t$. Let $t\uparrow L$; together with the reverse bound this provesThis is supremum convergence for decreasing continuous functions. The limit need not be continuous, so the conclusion does not assert the uniform convergence of the Dini theorem. It is precisely the interchange of these decreasing suprema and pointwise limit needed in the next part.