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 $\lim_{k\rightarrow\infty}p_{n_{k}}(z)=h(z)$ 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?
Best Answer
As per comments the crucial observation is that for every $n \ge 1$ we can find a polynomial $P_{n,h}$ st $|P_{n,h}(z)-h(z)| < 1/(2n), |z| \le n$ (eg some Taylor truncation of $h$).
But then since the closed disc is compact one can tweak the coefficients of $P_{n,h}$ very little and find $Q_{n,h}$ with rational coefficients st $|P_{n,h}(z)-Q_{n,h}(z)| < 1/(2n), |z| \le n$
Putting it together we get the sequence $Q_{n,h}$ satisfying $|h(z)-Q_{n,h}(z)| < 1/n, |z| \le n$. But now any compact $K$ is contained in $|z| \le n$ for $n \ge n_K$ so $Q_{n,h}$ converges uniformly on $K$ to $h$