Exercise 2, § 8

Let $(\varepsilon_n)$ be a strictly decreasing sequence of finite numbers > 0 which tend to 0. For each $x \in \mathbf{R}$, let $r_n(x)$ be the multiple of $\varepsilon_n$ which is the approximation by defect to $x$ to within $\varepsilon_n$. Then the sequence $(r_n(x))$ is increasing for all $x \in \mathbf{R}$ if and only if $\varepsilon_n$ is an integer multiple of $\varepsilon_{n+1}$ for each $n$.