$\sum_{n=0}^\infty f_n(z)$ is a holomorphic function in an open and connected set D. Prove that if $\sum_{n=0}^\infty \operatorname{Re} f_n(z)$ is uniformly convergent on any compact set in D, then $\sum_{n=0}^\infty \operatorname{Im} f_n(z)$ is either uniformly convergent on any compact set or divergent at any point in D.
This statement feels not so true. I don't understand how come $\sum_{n=0}^\infty \operatorname{Im} f_n(z)$ can diverge and yet $\sum_{n=0}^\infty f_n(z) = \sum_{n=0}^\infty \operatorname{Re} f_n(z) + i \operatorname{Im} f_n(z)$ still converges.