Let $X$ a Banach Space and $Y$ a Normed Space.
Consider the sequence $(f_n)_n^\infty$ in $C(X,Y) = \{f:X \to Y | f$ is continuos function $\}$ such that $sup\|f_n(x)\| \leq \infty$ $\forall x \in X$
Prove that:
$ \forall A \neq \emptyset$ open set of X, Exist $M \in \mathbb{R} $ and $O \subset A$ open set such that $sup\|f_n(x) \| \leq M $ $\forall x \in O$ and $\forall n\in N$
Some hints ?