(related to Proposition: Addition 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 addition of real numbers \(x+y=y+x\) for all \(x,y\in\mathbb R\) follows from the commutativity of adding rational Cauchy sequences. For all rational Cauchy sequences \((x_n)_{n\in\mathbb N}, (y_n)_{n\in\mathbb N}\in M\) we have \[\begin{array}{rcll} x+y&=&((x_n)_{n\in\mathbb N}+ I)+((y_n)_{n\in\mathbb N}+ I)&\text{by definition of real numbers}\\ &=&[(x_n)_{n\in\mathbb N}+ (y_n)_{n\in\mathbb N}]+ I&\text{by definition of adding real numbers}\\ &=&[(y_n)_{n\in\mathbb N}+(x_n)_{n\in\mathbb N}]+ I&\text{by commutativity of adding rational Cauchy sequences}\\ &=&((y_n)_{n\in\mathbb N}+ I)+((x_n)_{n\in\mathbb N}+ I)&\text{by definition of adding real numbers}\\ &=&y+x&\text{by definition of real numbers}\\ \end{array}\]
