If $A \subset E \subset \mathbb{R}$, then $A$ is dense in $E$ in case $E = \overline{A} \bigcap E$. Assume$\{f_n\}_{n=1}^\infty$ is a sequence of functions continuous on $E$ and converging uniformly on a set $A$ dense in $E$. Prove that $\{f_n\}_{n=1}^\infty$ converges uniformly on $E$.
For this I am confused on how I am using dense and continuity to find uniform convergence. My idea it that I am somehow fix an $x$ and show cauchy criterion for $f_n(x)$ and that for every $x \in E$ and $a \in A$, $|x-a|$ can be made small. Thus $|f_n(x) – f_m(x)|$ can be shown to be $< \epsilon$ by using the add and subtract trick.
Can someone help me refine this and explain why?
Best Answer
You just need to use the fact that if $A\subseteq E$ is dense in $E$ and $g$ is a continuous on $E$ function, then $\sup_{x\in A}g(x) = \sup_{x\in E}g(x)$. In your case you'd need to apply this to $g$ being a certain difference.