Let $(f_n)$ be a sequence of real valued continuous functions defined on a complete metric space $X$ such that $f(x) :=$ lim $f_n(x)$ exists for all $x \in X$.
these are problems.
(a) is there a nonempty open subset $O$ and a constant $M < \infty$ such that $\mid f_n(x) \mid < M$ for all $x \in O$ and for all $n \ge 1$?
(b) Given $\epsilon > 0$, is there a nonempty open subset $O$ and an integer $N$ such that $\mid f(x) - f_n(x) \mid < \epsilon$ for all $x \in O$ and for all $n \ge N$ ?
I can prove that $C(X)$ is complete metric space. but I don't know how to choose open set $O$. where do I have to start to solve these problems?