Definition of Real Numbers

Proposition: Definition of Real Numbers

Let \((x_n)_{n\in\mathbb N}\), \((y_n)_{n\in\mathbb N}\) be rational Cauchy sequences. We call them equivalent, if their difference is convergent to \(0\), formally

\[(x_n)_{n\in\mathbb N}\sim(y_n)_{n\in\mathbb N}\quad\Longleftrightarrow\quad \lim_{n\to\infty } y_n-x_n =0\]

The relation “\(\sim\)” defined above is an equivalence relation, i.e. for a given rational Cauchy sequence \((x_n)_{n\in\mathbb N}\) we can consider a whole set of rational Cauchy sequences \((y_n)_{n\in\mathbb N}\) equivalent to \((x_n)_{n\in\mathbb N}\):

\[x:=\{(y_n)_{n\in\mathbb N}\text{ is a rational Cauchy sequence},~ (y_n)_{n\in\mathbb N}\sim (x_n)_{n\in\mathbb N}\}\quad\quad ( * )\]

The set¹ \(x\) is called a real number, and the rational Cauchy sequence \((x_n)_{n\in\mathbb N}\) is called a representation² of the real number. The set of all real numbers is denoted by \(\mathbb R\).

For practical purposes, \( ( * ) \) it equivalent with the notation

\[x:=(x_n)_{n\in\mathbb N} + I, \]
where \(I\) is the set of all rational sequences convergent to \(0\).

¹ Please note that real numbers are in fact sets.

² This has very important practical consequences, in particular it means that the same real number can be represented in many ways, especially in any numeral system (e.g. decimal or binary).