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