This problem is from Greene and Krantz-Function Theory of One Complex Variable Chapter 3. I could not find any couterexample. Could anyone help? Thanks.
2026-03-28 16:05:21.1774713921
Unif convergence of sequences of infinitely diff complex functions does not imply uniform convergence of derivative on compact sets.
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Let $f_n(x)=\frac{1}{n} \sin(nx)$. Then $(f_n)$ converges uniformly on $ \mathbb R$ to $f=0$.
We have $f_n'(x)= \cos(nx)$. Hence $(f_n')$ does not converge uniformly on compact subsets of $ \mathbb R$.