# Constructing the reals from the integers

A map $f\colon\mathbb{Z}\longrightarrow\mathbb Z$ is called a quasi-homomorphism if the set$$\{f(m+n)-f(m)-f(n)\,|\,m,n\in\mathbb{Z}\}$$is bounded. Let $R$ be the set of these functions. Let's consider the binary relation $\sim$ in $R$ defined by$$f_1\sim f_2\iff\{f_1(n)-f_2(n)\,|\,n\in\mathbb{Z}\}\text{ is bounded.}$$Then $\sim$ is an equivalence relation and $R/\sim$ is simply the set of all real numbers. To be more precise, if $0$ is the equivalence class of the null function, if $1$ is the equivalence class of the identity, if $+$ is the operation induced by the sum and if $\times$ is the operation induced by the composition, then $R/\sim$ is a field which is isomorphic to $\mathbb R$.

I read this years ago and I would like to have a reference for it, in which the details are provided.

Note: If you see no connection between $\mathbb R$ and $R$, consider the map$$\begin{array}{ccc}\mathbb R&\longrightarrow&R\\x&\mapsto&\left[\begin{array}{ccc}\mathbb{Z}&\longrightarrow&\mathbb Z\\n&\mapsto&\lfloor nx\rfloor\end{array}\right].\end{array}$$