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.

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 $R$.

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.