Exercise 12, § 1

One says that a sequence $(x_n)$ of real numbers belonging to the interval $(0, 1)$ is equally distributed in this interval if, for every pair of numbers $\alpha, \beta$ such that $0 \leq \alpha \leq \beta \leq 1$ one has, denoting by $v_n(\alpha, \beta)$ the number of indices $i$ such that $1 \leq i \leq n$ and $\alpha \leq x_i \leq \beta$

Prove that, if the sequence $(x_n)$ is equally distributed, and if $f$ is a regulated function in $(0, 1)$, one has (reduce to the case where $f$ is a step function). Converse. Prove that, in order that the sequence $(x_n)$ be equally distributed, it is sufficient that relation (2) hold for every numerical function $f$ belonging to an everywhere dense set in the space of continuous numerical functions in $[0, 1]$, endowed with the topology of uniform convergence.