= Solution
Set $g_n=f-f_n$. These are nonnegative <continuous functions>, decrease pointwise, and converge pointwise to zero. For $\varepsilon>0$, the open sets $U_n=\{x:g_n(x)<\varepsilon\}$ increase and cover $K$. A finite subcover exists by compactness; because the sets are increasing, one $U_N$ already covers $K$. Hence $0\le g_n(x)<\varepsilon$ for every $x\in K$ and every $n\ge N$, and $\|f-f_n\|_\infty\le\varepsilon$. Therefore $\boxed{\|f_n-f\|_\infty\to0}$. This is <Dini's theorem>; continuity of the limit is essential to the openness used in the proof.
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