If $f\in C(U)$, then $f^{\epsilon }\to f$ uniformly on compact subsets of $U$

Question

If $f\in C(U)$, then $f^{\epsilon }\to f$ uniformly on compact subsets of $U$

my Question how he says that $f$ is uniformly on $W$ i am so learner and the only hope i learn sobolev spaces is MATHSSTACK....... so thanks to all

Answer 1

$W$ is contained in a compact contained in $U$ (see The meaning of notation $\subset\subset$ in complex analysis). And continuous in a compact implies uniformly continuous...