Sequence of real numbers which are distinct modulo 1. Let $\{x_n\}_{n\in\mathbb N}$ a sequence of real numbers which are distinct modulo 1. Are the sequences
$$\{\varepsilon x_n\}_{n\in\mathbb N} \ \ (\varepsilon\rightarrow 0), \ \ \ \left\{\frac{\varepsilon x_n}{\sin(x_n-x_m)}\right\}_{n\in\mathbb N} \ \ (\varepsilon\rightarrow 0), m\neq n$$
still composed by real numbers which are distinct modulo 1?

For any sequence of distinct reals, there are at most countably many $\epsilon$ such that $\epsilon x_n\equiv x_m\pmod 1$ for some $n,m$. This is because these epsilon are characterized by three integers $n,m,\epsilon(x_n-x_m)$. Hence for all sufficiently small positive $\epsilon$, the sequence $\{\epsilon x_n\}_{n\in\mathbb N}$ will be composed of real numbers distinct modulo $1$.
The sequence with $\sin(x_n-x_m)$ in the denominator may not even be a sequence of real numbers because $x_n-x_m$ may be a multiple of $\pi$. Nevertheless, if we ignore theses exceptions the argument of the above still applies.
