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
