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).