Entire function can be approximated uniformly by polynomials with rational coefficients on every compact set.

I am working on Stein Complex Analysis Chapter 2 Problem 5. The question has a hint that I cannot prove. Basically, the hint states as follows:

Let $p_{1}, p_{2},cdots$ denote an enumeration of the collection of polynomials whose coefficients have rational real and imaginary parts. Given $h$ entire, there exists a sequence ${n_{k}}$ such that $lim_{krightarrowinfty}p_{n_{k}}(z)=h(z)$ uniformly on every compact subset of $mathbb{C}$.

I understand that this is pretty similar to the Runge theorem:

Any function holomorphic in a neighborhood of a compact set $K$ can be approximated uniformly on $K$ by rational functions whose singularities are in $K^{c}=mathbb{C}setminus K$. If $K^{c}$ is connected, any function holomorphic in a neighborhood of $K$ can be approximated uniformly on $K$ by polynomials.

To prove the hint quickly, I tried to prove it using this theorem: as $h$ was entire, it is of course holomorphic in a neighborhood of any compact set $K$. Thus, it can be approximated by rational functions whose singularities are in $K^{c}$.

Then I got stuck. If $K^{c}$ is connected, then I can say $h(z)$ is approximated by a sequence of polynomial, and by density it is clear that this sequence of polynomial can be approximated by a sequence of polynomials with rational coefficients. Then we are done.

But generally speaking we do not have $K^{c}$ to be connected. What should I do?