# 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:

1. $$|x|=0$$ if and only if $$x=0$$ for all $$x\in\mathbb R$$.
2. $$|x-y|=|y-x|$$ for all $$x,y\in\mathbb R$$ (symmetry)
3. $$|x+y|\le |x| + |y|$$ 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.

Proofs: 1

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

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983