Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 21H / c / Solution

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