Uniform convergence in an infinite union of domain

Question

If a sequence of complex functions $\{f_k(z)\}$ converges uniformly on each of the domains $D_j $, let the union $D=D_1\cup D_2...\cup D_n$, if $n=\infty$, does sequence converge uniformly on the union?

For some finite number $n$, I think the answer is true, as could be shown using Cauchy's criterion for series. I'm not pretty sure how I could consider this infinite case.

Thanks for the help:)

Answer 1

Not true. Let $D_j=\{z: |z|<1-\frac 1 j\}$ and $f_k(z)=z^{k}$. Then $f_k\to 0 $ pointwise on $D =\cup D_n$ and it converges uniformly to $0$ on each $D_j$ but the convergence is not uniform on $D$ because $f_k(1-\frac 1 k) \to e^{-1}$.