Should a sequence of entire functions that converges on the disk to an entire function converge on the whole plane?

Question

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of entire functions and let $f$ be an entire function such that, for every compact subset $K$ of the unit disk, it holds that: $$\|f_n-f\|_{L^\infty(K)}\rightarrow0, n\rightarrow\infty.$$ Is it true that if $H$ is a compact subset of $\mathbb{C}$ then $$\|f_n-f\|_{L^\infty(H)}\rightarrow0, n\rightarrow\infty?$$ My guess is that the answer is no, and probably the counterexample is obvious, but I'm having hard time finding it.

Answer 1

$f_n(z)=(\frac z 2)^{n}$ is a counter-example. Here $f=0$ and you don't have convegrence when $H=\{3\}$.