Proof of power series uniform convergence on compact set I proved:
If a power series converges pointwise on a compact set then it converges uniformly. 
Please could somebody check my proof? 
My idea is to use Abel's theorem:
Let $g(x) = \sum_{n=0}^\infty a_n x^n$ be a power series that converges at the point $x=R > 0$. Then the series converges uniformly on $[0,R]$. A similar result holds for $x=-R$.
Let $K$ be a compact set. Let $M = \max K, m = \min K$. Let $p(x)=\sum_{n=0}^\infty a_n x^n$ be pointwise convergent on $K$. 
If $m < 0 < M$ then by Abel's theorem $p$ converges uniformly on $[0,M]$ and by the negative case uniformly on $[m,0]$ therefore uniformly on $K \subseteq [m,M]$. The argument is similar if $0 < m < M$ or $m < M < 0$.

Your proof is correct. Presentation may be improved by preceding it with a lemma: if a series converges uniformly on each of the sets $E_1,\dots,E_m$, then it converges uniformly on $\bigcup_{i=1}^m E_i$. (That is, uniformity of convergence is preserved under finite unions.)
Then you have $K\subseteq E_1\cup E_2$ where $E_1$ is the closed interval with endpoints $0,m$ and $E_2$ is the closed interval with endpoints $0,M$. 

Aside: I can't figure out if the statement remains true in the complex plane.
