# Proposition: Multiplication of Real Numbers

According the definition of real numbers, we can identify real numbers $$x,y \in \mathbb R$$ with equivalence classes $$x=(x_n)_{n\in\mathbb N}+ I$$ and $$y=(y_n)_{n\in\mathbb N}+ I$$ for some rational Cauchy sequences $$(x_n)_{n\in\mathbb N}$$ and $$(y_n)_{n\in\mathbb N}$$ and where $$I$$ denotes the set of all rational sequences, which converge to $$0$$.

Based on the multiplication of rational Cauchy sequences, we define a new multiplication operation "$$\cdot$$" for all real numbers by setting

$\begin{array}{ccl} x\cdot y=((x_n)_{n\in\mathbb N} + I)\cdot ((y_n)_{n\in\mathbb N} + I)=(x_n \cdot y_n)_{n\in\mathbb N} + I, \end{array}$

where $$(x_n \cdot y_n)_{n\in\mathbb N} + I$$ is also a real number, called the product of the real numbers $$x$$ and $$y$$. The product exists and is well-defined, i.e. it does not depend on the specific representatives $$(x_n)_{n\in\mathbb N}$$ and $$(y_n)_{n\in\mathbb N}$$ of $$x$$ and $$y$$.

Proofs: 1

Definitions: 1 2 3
Parts: 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Propositions: 20 21 22 23 24 25 26 27 28 29 30

Github: ### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013