# Proof

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.

Github: ### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013