Let $(X,d)$ a complete metric space, $f_n : X \rightarrow \mathbb{R}$ a sequence of continuous functions and $f$ its pointwise limit. Prove that: a) exists an open dense set $W$ such that $f$ is locally bounded in $W$. b) the set where $f$ is continuous is a $G_\delta$ in $X$.
I think $W$ should be define this way (for part a) ) $W=\bigcup_{k\ge 1} int(\bigcap_{n \ge 1} \left \{\mid f_n\mid\le k \right \}) $, but I'm not sure how to prove it's dense. Any help? Thanks