Definition:Real Number/Cauchy Sequences

Definition
Consider the set of rational numbers, $\Q$.

For any two Cauchy sequences of rational numbers $X = \sequence {x_n}, Y = \sequence {y_n}$, define an equivalence relation between the two as:


 * $X \equiv Y \iff \forall \epsilon \in \Q_{>0}: \exists n \in \N: \forall i, j > n: \size {x_i - y_j} < \epsilon$

A real number is an equivalence class $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.

The set of real numbers is denoted $\R$.

Also see

 * Equivalence Relation on Cauchy Sequences, which justifies the construction