For holomorphic function on an open set, does there exists a sequence of polynomials that converges uniformly to it on any compact subset? I was trying to give a proof or counterexample of the following statement: For holomorphic function $f$ on an open set $\Omega$, there exists a sequence of polynomials $(p_n)_{n\geq1}$ that converges uniformly to $f$ on a compact subset $S\subset\Omega$.
I believe this statement is not true. I was thinking the following case: Let $f(z)=1/z$, $\Omega=\mathbb{C}\setminus\{0\}$ and $S$ to be an annulus with the inner boudary really close to origin, like $S=\{z\in \mathbb{C}:a\leq|z|\leq 1\}$ for some very small $a$. Then I stuck at this point. Am I on the right track?
Thanks for any help.

Yes, you're definitely on the right track. Now integrate $1/z$ on a circle centered at $0$ and also integrate the polynomials that supposedly uniformly approximate it on said compact circle.
It's sort of an interesting question because it depends crucially on the topology of $\Omega$: the answer is yes if and only if $\Omega$ is simply connected. 
