= 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 $L\geq0$. Since $f\leq f_n$, certainly $\sup_K f\leq L$.
For any $t<L$, the sets $E_n=\{x\in K:f_n(x)\geq t\}$ 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 $f\geq t$, so $\sup_K f\geq t$. Let $t\uparrow L$; together with the reverse bound this proves
$$
\boxed{\sup_K f_n\longrightarrow\sup_K f.}
$$
This 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.
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