Basic Analysis I — Real Analysis · Problem 6.1.6

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-convergence-products-counterexample

Problem

Find an example of a sequence of functions {fn}n=1∞\{f_n\}_{n=1}^\infty and {gn}n=1∞\{g_n\}_{n=1}^\infty that converge uniformly to some ff and gg on some set AA, but such that {fngn}n=1∞\{f_n g_n\}_{n=1}^\infty (the multiple) does not converge uniformly to fgfg on AA. Hint: Let A:=RA \mathrel{:=} \mathbb{R}, let f(x):=g(x):=xf(x) \mathrel{:=} g(x) \mathrel{:=} x. You can even pick fn=gnf_n = g_n.

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