Rigorous Proof of Infinite Number of Equivalence Classes

Question

Let X $=\mathbb{R}$

Let $x \sim y \leftrightarrow x - y \in \mathbb{Z}$.

It is intuitively obvious why this would have an inifinite number of equivalence classes. Is there a rigorous way of proving this?

Answer 1

Say $x,y\in [0,1)$ with $x\neq y$. Can $x-y\in\mathbb{Z}$?