Basic Analysis I — Real Analysis · Problem 6.1.8

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-convergence-squeeze-theorem

Problem

Let {fn}n=1∞\{f_n\}_{n=1}^\infty, {gn}n=1∞\{g_n\}_{n=1}^\infty and {hn}n=1∞\{h_n\}_{n=1}^\infty be sequences of functions on [a,b][a, b]. Suppose {fn}n=1∞\{f_n\}_{n=1}^\infty and {hn}n=1∞\{h_n\}_{n=1}^\infty converge uniformly to some function f:[a,b]→Rf : [a, b] \to \mathbb{R} and suppose fn(x)≤gn(x)≤hn(x)f_n(x) \leq g_n(x) \leq h_n(x) for all x∈[a,b]x \in [a, b]. Show that {gn}n=1∞\{g_n\}_{n=1}^\infty converges uniformly to ff.

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