Proving that if $\sum_0^\infty f_n$ converges uniformly on $D$ to f and $f_n$ is bounded, then f is bounded. "Prove that if each $f_n$ is a bounded function and $\sum_0^\infty f_n$ converges uniformly on $D$ to $f$, then
$f$ is a bounded function"
I don't know how to do this at all. Any help appreciated. 

Let $\epsilon \gt 0$ be given. $f_n$ converges uniformly to $f$ so $\exists N \in \mathbb{N}$ s.t.
$|f_n - f| \lt \epsilon$ whenever $n \geq N$.
For a fixed $n = N$,
$|f_N - f| \lt \epsilon \implies$
$-\epsilon + f_N \lt f \lt \epsilon + f_N$ --- $(1)$
But $f_N$ is bounded so $|f_N| \leq M_N$ (for some $M_N \gt 0$) 
$\implies -M_N \leq f_N \leq M_N$
So $-\epsilon + f_N \geq -\epsilon - M_N$ and 
$\epsilon + f_N \leq \epsilon + M_N$
So $(1)$ becomes $-\epsilon - M_N \lt f \lt \epsilon + M_N$
$\implies |f| \lt \epsilon + M_N$ so $f$ is bounded.
