Let $\{r_1, \dots ,r_n , \dots\}$ be an enumeration of the rationals in the interval $[0, 1]$. Define, for $n\in N$ and for each $x ∈ [0, 1]$, $$f_n(x)=\left\{\begin{array}{ll}1&x \in \{r_1, · · · ,r_n\}\\ 0& \mbox{otherwise,}\end{array}\right.$$ then is $f_n$ Riemann integrable on $[0, 1]$ for each $n$?
My work: For any partition $P$ of $[0, 1]$, the subintervals contains the points $r_1,···,r_n$ thus the Riemann upper sum $U(P,f_n)$ and lower sum $L(P,f_n)$ are different.
However, can we say $\inf_PU(P,f_n)=\sup_PL(P,f_n)$ ?