For some Banachspace $A$ we have a sequence of continuous functions $g_n:A\rightarrow \mathbb{R}$ pointwise converging to some $g:A\rightarrow\mathbb{R}$. Prove that for any $\epsilon>0$ there exist $\emptyset\not=U\subset A$ open and $N\in\mathbb{N}$ such that for all $n>N$ we have $\sup_{x\in U}\left|g_n(x)-g(x)\right|<\epsilon$.
I'm not sure how to approach this problem. Is it a good idea to prove something like local boundedness first?