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

(b) Let $f_{n}: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function for each $n=1,2,3, \ldots$ and suppose that there is a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f_{n}(x) \rightarrow f(x)$ for each $x \in \mathbf{R}$. Prove that for each $\epsilon>0$, there exists an integer $N_{0}$ and a non-empty open interval $I \subset \mathbf{R}$ such that $\left|f_{n}(x)-f(x)\right| \leqslant \epsilon$ for all $n \geqslant N_{0}$ and $x \in I$.

[Hint: consider, for $N=1,2,3, \ldots$, the sets

$$\left.Q_{N}=\left\{x \in \mathbf{R}:\left|f_{n}(x)-f_{m}(x)\right| \leqslant \epsilon: \forall n, m \geqslant N\right\} .\right]$$