Prove that $f_n = \sum_{k=0}^n 1_{\{\varphi(k)\}}$ for $\varphi = \mathbb{N} \rightarrow \mathbb{Q} \cap [0,1]$ is Riemann integrable

Question

We consider a bijection $\varphi = \mathbb{N} \rightarrow \mathbb{Q} \cap [0,1]$.

We consider also de sequence of functions $(f_n)_{n \in \mathbb{N}}$ defined on $[0,1]$ by $f_n = \sum_{k=0}^n 1_{\{\varphi(k)\}}$.

How to prove that the functions $f_n$ are Riemann-integrables ?

My idea is to prove that $m_i = \inf_{x \in [x_{i-1}-x_i]} f_n(x)$ and $M_i = \sup_{x \in [x_{i-1}-x_i]} f_n(x)$ are equal, for $x_0 = 0 < x_1 < ... < x_{n-1} < x_n = 1$ a subdivision of $[0,1]$ but I don't see how to do it.

Could someone help me ?

Answer 1

Finite sum of Riemann integrable functions is still Riemann integrable, so it suffices to consider $1_{\{\varphi(k)\}}$, but this is clear.