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Proposition: The distance of real numbers makes real numbers a metric space.
Let \(x,y\) be elements of the field of real numbers \((\mathbb R, + ,\cdot)\). The absolute value of the difference \(|x-y|\) of two real numbers \(x\) and \(y\) is called their distance. This distance defines a metric on \(\mathbb R\).
In other words, \((\mathbb R,|~|)\) is a metric space and the distance \(|~|\) fulfills the following properties:
- \(|x|=0\) if and only if \(x=0\) for all \(x\in\mathbb R\).
- \(|x-y|=|y-x|\) for all \(x,y\in\mathbb R\) (symmetry)
- \(|x+y|\le |x| + |y|\) for all \(x,y\in\mathbb R\) (triangle inequality)
- A generalization of this proposition can be formulated also for the complex numbers.
- Because \((\mathbb R,|~|)\) is a metric space and \((\mathbb R, + ,\cdot)\) is a field, we also say, that the field of real numbers is a valuated field.
Table of Contents
Definitions: 2 3 4
Proofs: 6 7 8 9 10 11 12 13 14 15 16 17
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983