Denote by $\mathbb{D}$ the open unit disk of the complex plane $\mathbb{C}$ and let $D = \{z_n\}_{n\geq 1}$ be a dense set on $\mathbb{D}$. I'm looking for a sequence of analytic functions $(f_n)$ on $\mathbb{C}$ such that converges uniformly to an analytic function $f$ on $\mathbb{C}\smallsetminus\overline{\mathbb{D}}$, converges pointwise on the dense set $D$ but does not converge on $\mathbb{D}\smallsetminus D.$ Can we construct such an example?
Thank you very much!
No, if $f_n$ converges uniformly on $|z|=2$ then it does so on $|z|\le 2$.
$$\sup_{|z|\le 2} |f_n(z)-f_m(z)|\le \sup_{|z|=2} |f_n(z)-f_m(z)|$$