◀ ▲ ▶Branches / Analysis / Proposition: The distance of real numbers makes real numbers a metric space.
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 \(xy\) 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 \(x\in\mathbb R\).
 \(xy=yx\) for all \(x,y\in\mathbb R\) (symmetry)
 \(x+y\le x + y\) for all \(x,y\in\mathbb R\) (triangle inequality)
Notes
 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
Proofs: 1
Mentioned in:
Corollaries: 1
Definitions: 2 3 4
Explanations: 5
Proofs: 6 7 8 9 10 11 12 13 14 15 16 17
Propositions: 18
Theorems: 19
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983