Theorem
The uniform limit of continuous functions is continuous. More precisely, let
$ (f_n)$ be a sequence of functions on a set $ S \subseteq \mathbf{R}, $
suppose $f_n \longrightarrow f$ uniformly on $S$, and suppose $S = dom(f).$
If each $f_n$ is continuous on $S$, then $f$ is continuous on $S$.
I would like to prove that some $f_n$ is not uniformly continuous. So my question is, if I were to set this up like a proof by contradiction or a proof by contra-positive, would that be sufficient?
i.e. If I wrote: Assume, to get a contradiction, that $f_n \longrightarrow f$ uniformly on $S=dom(f)$ and each $f_n$ is continuous on $S$, then $f$ is not continuous on $S$ .....or something like this?
Is this structure sufficient to contradict "$f_n \longrightarrow f$ uniformly on $S$"