# 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 $$0$$.

Proofs: 1 Explanations: 1

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### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013

#### Footnotes

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).