Help with counter-example on uniformly convergence.

Question

Show that the sequence: $$h_n(x)=\begin{cases}\frac{x}{n}\left(1+\frac{1}{n}\right) \mbox{ if } x=0\mbox{ or }x\notin\mathbb{Q}\\ x(b+\frac{1}{n})(1+\frac{1}{n})\mbox{ if } x\in\mathbb{Q}. \end{cases}$$

Does not uniformly converges at any bounded interval.

Answer 1

This is false. On any bounded interval, $h_n$ converges uniformly to $bx\cdot\chi_{\mathbb Q}(x).$