# 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.

## Notation

While the symbol $\R$ is the current standard symbol used to denote the **set of real numbers**, variants are commonly seen.

For example: $\mathbf R$, $\RR$ and $\mathfrak R$, or even just $R$.

## Also known as

When the term **number** is used in general discourse, it is often tacitly understood as meaning **real number**.

They are sometimes referred to in the pedagogical context as **ordinary numbers**, so as to distinguish them from **complex numbers**

However, depending on the context, the word **number** may also be taken to mean **integer** or **natural number**.

Hence it is wise to be specific.

## Also see

- Equivalence Relation on Cauchy Sequences, which justifies the construction

- Results about
**real numbers**can be found here.