(a) State the Baire category theorem, in its closed-sets version.

(b) For every $n \in \mathbb{N}$ let $f_{n}$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$, and let $g(x)=1$ when $x$ is rational and 0 otherwise. For each $N \in \mathbb{N}$, let

$$F_{N}=\left\{x \in \mathbb{R}: \forall n \geqslant N \quad f_{n}(x) \leqslant \frac{1}{3} \quad \text { or } \quad f_{n}(x) \geqslant \frac{2}{3}\right\} .$$

By applying the Baire category theorem, prove that the functions $f_{n}$ cannot converge pointwise to $g$. (That is, it is not the case that $f_{n}(x) \rightarrow g(x)$ for every $x \in \mathbb{R}$.)