(related to Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication)
According to the definition of real numbers, a real number \(x\) is the equivalence class of all rational Cauchy sequences, the difference of which converges to \(0\in\mathbb Q\).
We have proven that the set of all rational Cauchy sequences, together with the addition of rational Cauchy sequences "\(\oplus\)" and the multiplication of rational Cauchy sequences "\(\odot\)", forms the commutative unit ring of all rational Cauchy sequences \((M , \oplus , \odot)\). We have also proven that the set \(I:=\{(a_n)_{n\in\mathbb N}~|~a_n\in\mathbb Q,\lim a_n=0\}\) i.e. the set of all rational sequences, which converge to \(0\), is an ideal of \(M\), formally \(I\lhd M\).
With these prerequisites, we can use a corresponding lemma about factor rings to conclude the existence of a commutative factor ring \((M/I , + , \cdot)\), in which "\(+\)" and \(\cdot\) denote new addition (respectively multiplication) operations for the factor ring \(M/I\).
To identify the factor ring \((M/I , + , \cdot)\) with the field of real numbers \((\mathbb R , + , \cdot)\), it remains to be shown that each equivalence class \((x_n)_{n\in\mathbb N} + I\) with \((x_n)_{n\in\mathbb N} + I\neq (0)_{n\in\mathbb N}+I\) has a multiplicative inverse element in \((M/I , + , \cdot)\). This is equivalent with the existence of inverse elements with respect to multiplication.