Basic Analysis I — Real Analysis · Problem 6.2.10

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · dinis-theorem

Problem

(Hard): Prove Dini's theorem: Let fn:[a,b]→Rf_n : [a, b] \to \mathbb{R} be a sequence of continuous functions such that 0≤fn+1(x)≤fn(x)≤⋯≤f1(x)for all n∈N.0 \leq f_{n+1}(x) \leq f_n(x) \leq \cdots \leq f_1(x) \quad \text{for all } n \in \mathbb{N}. Suppose {fn}n=1∞\{f_n\}_{n=1}^\infty converges pointwise to 00. Show that {fn}n=1∞\{f_n\}_{n=1}^\infty converges to zero uniformly.

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