A Construction of the Real Numbers

Considering $\mathcal{C}_\mathbb{Q}$ as the set of all Cauchy sequences of rational numbers, we define a relation on $\mathcal{C}_\mathbb{Q}$:

This relation is an equivalence relation, as it satisfies:

1. Reflexivity: $\lim_{n \to \infty} (a_n - a_n) = 0$.

2. Symmetry: $\lim_{n \to \infty} (a_n - b_n) = 0$ implies $\lim_{n \to \infty} (b_n - a_n) = 0$.

3. Transitivity: If $\lim_{n \to \infty} (a_n - b_n) = 0$ and $\lim_{n \to \infty} (b_n - c_n) = 0$, then $\lim_{n \to \infty} (a_n - c_n) = 0$.

For any $k \in \mathbb{Q}$, we consider the sequence $(k)_{n \in \mathbb{N}}$, which is a Cauchy sequence and forms an equivalence class. Thus, we identify each rational number with its equivalence class:

We can extend this to irrational numbers by considering equivalence classes where the limit is not rational, considering $i \notin \mathbb{Q}$, then we have:

Therefore, the set of real numbers can be defined as the set of all equivalence classes of Cauchy sequences of rational numbers:

This construction of the real numbers ensures a bijection between $\mathbb{R}$ and the set of all Cauchy sequences of rational numbers, providing a complete model of $\mathbb{R}$ from $\mathbb{Q}$.