Pointwise convergence of this function
Solution 1:
Hint
Let $ q $ be an integer $\ge 1 $. Then
$$n\ge q \implies \frac 1q\in\{1,\frac 12,\cdots, \frac 1n\}$$ thus $$n\ge q\implies f_n(\frac 1q)=\frac 1q$$ and $$\lim_{n\to+\infty}f_n(\frac 1q)=\frac 1q$$