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## 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).

## 12.Lemma: Unit Ring of All Rational Cauchy Sequences

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### Bibliography (further reading)

 Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013

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