Consider a sequence of holomorphic function $\{f_n\}$ on a domain $D$ with limit function $f$ . It is known that if the convergence is compact (i.e. converge uniformly to $f$ in every compact subset of $D$), $f$ is holomorphic on $D$.
Then is there an example that $f$ is holomorphic on $D$ but the convergence is not compact?
Edit: Here is a similar question but this doesn't answer to my question since I require $f$ to be holomorphic. So this is not the duplicate!