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 subtraction of real numbers, written \(x-y\), is defined as the addition of the first real number \(x\) with the inverse of the second real number with respect to addition \((-y)\), formally \[\begin{array}{rcl}x-y&=&((x_n)_{n\in\mathbb N} + I)-((y_n)_{n\in\mathbb N} + I)\\ &=&((x_n)_{n\in\mathbb N} + I)+((-y_n)_{n\in\mathbb N} + I)\\ &=&(x_n-y_n)_{n\in\mathbb N} + I\\ &=&x+(-y).\end{array}\]
The result of the subtraction is called the difference of the real numbers $x$ and $y$.
Definitions: 1 2
Parts: 3
Proofs: 4 5 6 7 8
Propositions: 9 10 11
