How can we prove that the sequence of functions defined as $$ f_n(x) = \begin{cases} 1/n, & x \in\mathbb{Q} \\ 1/n+ 1, & x \in \mathbb{R}\setminus \mathbb{Q} \end{cases} $$
is pointwise convergent on $\mathbb{R}$ and not uniformly convergent on any interval of positive length of $\mathbb{R}$ ?
We cannot, since it is not true. Your sequence of functions converges uniformly to$$\begin{array}{rccc}f\colon&\mathbb R&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}0&\text{ if }x\in\mathbb Q\\1&\text{ if }x\in\mathbb R\setminus\mathbb Q\end{cases}\end{array}$$on $\mathbb R$.