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