Definition: In a natural construction for the real numbers of Norbert A’Campo, a real number is an equivalence class of slopes. A slope is by definition a map $\lambda: \mathbb{Z} \rightarrow \mathbb{Z}$, with the property that the set $\{\lambda(m + n) - \lambda(m) - \lambda(n) \mid m,n \in \mathbb{Z} \}$ is finite. For all $p,q \in \mathbb{Z}$ and $q > 0$, let the map $\lambda^p_q: \mathbb{Z} \rightarrow \mathbb{Z}$ be defined by $\lambda^p_q(n) := \min \{k \in \mathbb{N} \mid pn \leq qk\}$ for all $n \in \mathbb{N}$ and $\lambda^p_q(−n) = -\lambda^p_q(n)$ for all $n \in \mathbb{Z}$ where $n < 0$. To prove that $\lambda^p_q$ is a slope, we have to show the set $\{\lambda^p_q (m + n) + [-\lambda^p_q (m)] + [-\lambda^p_q (n)] \in \mathbb{Z} \mid m,n \in \mathbb{Z}\}$ is finite.
My attempt: in case of $m < 0$ and $n \geq 0$:
$\begin{array}{ll} & \{\lambda^p_q (m + n) - \lambda^p_q (m) - \lambda^p_q (n) \in \mathbb{Z} \mid n \in \mathbb{N} \wedge m \in \mathbb{Z} \backslash \mathbb{N} \} \\ = & \{\lambda^p_q (-m + n) + \lambda^p_q (m) - \lambda^p_q (n) \in \mathbb{Z} \mid m,n \in \mathbb{N}\} \end{array}$
- My problem: From $m,n \in \mathbb{N}$, we have $pm < 0$ and $pn < 0$ for all $p < 0$. Therefore, $\lambda^p_q (m) := \min \{i \in \mathbb{N} \mid p \cdot m \leq q \cdot i \} = 0$ and $\lambda^p_q (n) := \min \{j \in \mathbb{N} \mid p \cdot n \leq q \cdot j \} = 0$. Hence, $\{\lambda^p_q (-m + n)\} = \{\min \{k \in \mathbb{N} \mid p \cdot (-m + n) \leq q \cdot k \} \mid m,n \in \mathbb{N} \}$ is not finite.
I think that I was wrong at some where but I can not find. Hope your helps.