This is a problem in PMA Rudin p.165
Let $C_m=(-\frac{1}{m^2},-\frac{1}{(m+1)^2}), \forall m\in\mathbb{N}$
Define $f_n=\frac{1}{1+n^2x}$ on its natural domain,$\forall n\in \mathbb{N}$.
Define $f(x)=\sum_{n=1}^{\infty} f_n(x)$ where $x\in \mathbb{R}\setminus (\{0\} \cup \{-\frac{1}{n^2}\in \mathbb{R}|n\in\mathbb{N}\})$.
Then $f$ is well-defined.
I have shown that $\sum f_n\rightarrow f$ uniformly on $C_m$ for any $m\in\mathbb{N}$.
Now, i don't know how to prove that $\sum f_n\rightarrow f$ uniformly on $f$'s natural domain. Is there a theorem for this?