Runge uniform convergence theorem on closed curves

Question

I just came across this statement attributed to Runge (1885, in an on-line preview of Remmert's "Classical Topics in Complex Function Theory"): If an expression of the form lim $g_n(x)$ converges uniformly on a closed curve of finite length, then it also converges uniformly in its interior. However, so far I had no success in locating a proof of this statement, neither in my collection of books in complex analysis nor by searching the web. I will appreciate any suggestion. Thanks.

Answer 1

To 'prove' something, we need to 'state' the proposition. The sentence you quote is not false but for it to be true, it should be restated carefully. Let me formulate the proposition.

Proposition Consider a region(open connedted set) $D \subseteq \mathbb{C}$ whose boundary $C$ is a closed curve and a sequence of holomorphic functions $g_n:D \longrightarrow \mathbb{C}$ which are continuous on the closure of $D$. Suppose that $g_n$ converges uniformly on $C$. Then $g_n$ converges uniformly in the interior of $C$ to a holomorphic function.

I guess the above is enough for most purposes where the statements like quoted one is required.

The proof is simple and just a direct consquence of the Cauchy integral formula as mentioned by Conrad. You may visit the following link to find the proof.

The proof of the proposition