Let \((X,d)\) be a compact metric space. Suppose that \(f_n \colon X \to [0,\infty)\) is a sequence of continuous functions with \(f_n(x) \geq f_{n+1}(x)\) for all \(n \in \mathbb{N}\) and \(x \in X\), and such that \(f_n \to 0\) pointwise on \(X\). Prove that \(\{f_n\}_{n=1}^\infty\) converges uniformly on \(X\).
Solution
1. Reduce to Dini’s theorem.
this is Dini’s theorem: a monotone pointwise-convergent sequence of continuous functions on a compact space converges uniformly.
2. For \(\eps > 0\) and each \(n\), the set \(U_n \da \{x : f_n(x) < \eps\}\) is open.
\(f_n\) is continuous, and \(U_n = f_n^{-1}(-\infty, \eps)\) is the preimage of an open set.
3. The sets \(U_n\) increase and cover \(X\).
\(f_n \ge f_{n+1}\) gives \(U_n \subseteq U_{n+1}\).
4. Some \(U_N\) already equals \(X\).
\(\{U_n\}\) is an open cover of the compact space \(X\) (<1>2, <1>3), so it has a finite subcover \(U_{n_1} \subseteq \cdots \subseteq U_{n_k}\); since the \(U_n\) are nested, \(X = U_{n_k}\), and taking \(N = n_k\) works.
5. Uniform convergence: \(f_n \to 0\) uniformly on \(X\).
by 4, for \(\eps > 0\) there is \(N\) with \(f_N(x) < \eps\) for all \(x\); monotonicity gives \(0 \le f_n(x) \le f_N(x) < \eps\) for all \(n \ge N\) and all \(x\).
6. Q.E.D.