# On Compactness in Runges theorem

Let $f$ be holomorphic function in an open set $\Omega$ in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions, converging uniformly to $f$ on $\Omega$.

For each $f_n$, let $\{{g_{n,m}}\}_{m=1}^{\infty}$ be a sequence of polynomials which converge uniformly to $f_n$ on $\Omega$.

Question: Does it follows that $f$ is a uniform limit on $\Omega$ of a sequence of polynomials? (More precisely, here do we need to assume $\Omega$ is compact set?)

Such arguments come in Runge's approximations theorem, which I didn't find rigorously explained in many (any) book. If one knows any elementary exposition on Runge's theorem, I would like to see it.)

• I think you mean that $g_{n,m}\to f_{\color{red}n}$ uniformly as $m\to\infty$ for each $n$, right? – sranthrop Mar 12 '15 at 12:44
• Oh yes! Thanks for notification. – Groups Mar 13 '15 at 6:21

Fix $\varepsilon>0$. For each $n\in\mathbb N$ there is some $m(n)\in\mathbb N$ such that $||g_{n,m(n)}-f_n||<\varepsilon/2$. Moreover, there is some $N\in\mathbb N$ such that $||f_n-f||<\varepsilon/2$ for each $n\geq N$. Finally, $||g_{n,m(n)}-f||\leq||g_{n,m(n)}-f_n||+||f_n-f||<\varepsilon$ for each $n\geq N$. Here, $||\cdot||$ is the sup-norm on $\Omega$. This shows, that the sequence $g_{n,m(n)}$ of polynomials converges uniformly to $f$ on $\Omega$.