Let $ \{r_1, r_2 ,r_3,... \}$ be an enumeration of $\mathbb{Q}$. For each $r_n \in \mathbb{Q}$ define:
$$u_n(x)=\begin{cases} 1/{2^n} & x>r_n \\ 0 & x \leq r_n \end{cases} $$
and let $$h (x) = \sum _{n=1}^{\infty} u_n (x) $$ Where does this function converge to?
I know that $|u_n (x)| \leq 1/2^n$ and thus by the M-test the series converges uniformly but although it might be trivial I cannot see where. Is it something I am missing?
Thank you.