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).
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| created: 2015-01-25 09:02:17 | modified: 2020-07-04 11:01:28 | by: bookofproofs | references: [696]
[696] Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013