Let be $f_n:\mathbb{R}\rightarrow[0, +\infty)$ as sequence of continuous and non-negative functions. Can $\{f_n\}_{n\geq1}$ exist such that: $$\{\ x \in \mathbb{R}\ |\ \sup_{n\geq1}f_n(x)=+\infty\}=\mathbb{Q}?$$ How can I prove that this statement is either false or true by using the Baire's category theorem and knowing that $\mathbb{Q}$ is a $F_\sigma$ set?
I wanted to prove that the set $\{\ x \in \mathbb{R}\ |\ \sup_{n\geq1}f_n(x)=+\infty\}$ is a $G_\delta$ but I don't know how.
We have $$ \sup_{n\geq1}f_n(x)=+\infty \iff \forall k \in \Bbb N: \sup_{n\geq1}f_n(x) > k\\ \iff \forall k \in \Bbb N: \exists n \in \Bbb N: f_n(x) > k $$ and therefore $$ \{\ x \in \mathbb{R} \mid \sup_{n\geq1}f_n(x)=+\infty\} = \bigcap_{k\ge 1} \bigcup_{n\ge 1} \{ x \in \Bbb R \mid f_n(x) > k\} $$ and that is a $G_\delta$ set.