(related to Proposition: Multiplication of Real Numbers Is Commutative)
Let \(x\) and \(y\) be real numbers, which by definition means that they are the equivalence classes \[\begin{array}{rcl}x&:=&(x_n)_{n\in\mathbb N} + I,\\ y&:=&(y_n)_{n\in\mathbb N} + I.\end{array}\] In the above definition, \((x_n)_{n\in\mathbb N}\) and \((y_n)_{n\in\mathbb N}\) denote elements of the set \(M\) of all rational Cauchy sequences, which represent the real numbers \(x\) and \(y\), while \(I\) denotes the set of all rational sequences, which converge to \(0\).
The commutativity of the multiplication of real numbers \(x\cdot y=y\cdot x\) for all \(x,y\in\mathbb R\) follows from the commutativity of multiplying rational Cauchy sequences. For all rational Cauchy sequences \((x_n)_{n\in\mathbb N}, (y_n)_{n\in\mathbb N}\in M\) we have \[\begin{array}{rcll} x\cdot y&=&((x_n)_{n\in\mathbb N}+ I)\cdot ((y_n)_{n\in\mathbb N}+ I)&\text{by definition of real numbers}\\ &=&[(x_n)_{n\in\mathbb N}\cdot (y_n)_{n\in\mathbb N}]+ I&\text{by definition of multiplying real numbers}\\ &=&[(y_n)_{n\in\mathbb N}\cdot (x_n)_{n\in\mathbb N}]+ I&\text{by commutativity of multiplying rational Cauchy sequences}\\ &=&((y_n)_{n\in\mathbb N}+ I)\cdot ((x_n)_{n\in\mathbb N}+ I)&\text{by definition of multiplying real numbers}\\ &=&y\cdot x&\text{by definition of real numbers}\\ \end{array}\]
