Basic Analysis I — Real Analysis · Problem 6.2.11

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · monotone-pointwise-uniform-convergence

Problem

Suppose fn:[a,b]→Rf_n : [a, b] \to \mathbb{R} is a sequence of continuous functions that converges pointwise to a continuous f:[a,b]→Rf : [a, b] \to \mathbb{R}. Suppose that for every x∈[a,b]x \in [a, b], the sequence {∣fn(x)−f(x)∣}n=1∞\left\{\left|f_n(x) - f(x)\right|\right\}_{n=1}^\infty is monotone. Show that the sequence {fn}n=1∞\{f_n\}_{n=1}^\infty converges uniformly.

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