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I am trying to show and disprove uniform convergence in the following example:
$D=\{z\in \C | |z| < 1 \} \ and \ f_{n}:D\rightarrow \C : f_{n}(z)=\frac{1}{1+z^{n}} $
Proposition 1: $f_{n}$ converges uniformly in all $B(0)$ with $0
Proof 1: With $f_{n}(z)= (1+z^n)^{-1}$ put
f'_{n}(z)=-nz^{n-1}(1+z^n)^{-2}=0 \Rightarrow z=0
(I wanted to use that \lim sup |f_{n} - f | = 0 but I fail at finding the supremum of $|f_{n}-f|$) How does one find the supremum?
So I try finding an estimate instead: $|f_{n} - f| = |\frac{1}{1+z^{n}}-1| = |\frac{-z^{n}}{1+z^{n}}| \le \frac{r^{n}}{|1-r^{n}|} =: \epsilon$
Is this done correctly?
Proposition 2: $f_{n}$ does not converge uniformly in D
Proof 2: How can one show that something does not converge uniformly??
Thanks for suggestions.