I have read the following theorem in Complex Analysis by Cufi and Bruna on Page 420:
Runge's theorem for open sets: An open set $U$ has the property that every holomorphic function on $U$ is the uniform limit on compact sets of $U$ of a sequence of polynomials, if and only if $\mathbb C \setminus U$ has no bounded connected component.
Now I came across the following question:
For which among the following functions $f(z)$ defined on $G=\mathbb C \setminus \{0\}$, is there no sequence of polynomials approximating $f(z)$ uniformly on compact subsets of $G$?
$(a)\ e^z \\(b)\ \frac1z\\(c)\ z^2\\(d)\ \frac1{z^2}$
The answer according to me should be $(b)$ and $(d)$.
But if I try to apply the above theorem, I see that all the functions are holomorphic on $G$(open) and $\mathbb C \setminus G=\{0\}$ which has a bounded connected component, namely $\{0\}$. So according to the theorem, the correct answer should be $(a),(b),(c)$ and $(d)$.
Where am I going wrong in applying the theorem?
Why should (a), say, be the correct answer? We are working here with $\mathbb{C}\setminus\{0\}$, indeed, but what you can deduce from Runge's theorem is that there is some holomorphic function $f$ defined on that set for which there is no sequence of polynomials approximating $f$ uniformly on compact subsets of $\mathbb{C}\setminus\{0\}$, not that that's true for all holomorphic functions defined there.