\begin{definition}[Definition:Real Number/Cauchy Sequences]
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.
\end{definition}