# Proposition: Addition of Real Numbers

Any real number $$x\in\mathbb R$$ is by the corresponding proposition an equivalence class $$x:=(x_n)_{n\in\mathbb N} + I,$$ where $$(x_n)_{n\in\mathbb N}$$ denotes a rational Cauchy sequence representing the real number $$x$$, and where $$I$$ denotes the set of all rational sequences, which converge to $$0$$.

Given two real numbers $$x=(x_n)_{n\in\mathbb N}+ I$$ and $$y=(y_n)_{n\in\mathbb N}+ I$$, the addition of real numbers is defined by

$\begin{array}{rcccl} x+y&:=&(x_n+y_n)_{n\in\mathbb N}+ I, \end{array}$

where the result is the real number $$(x_n + y_n)_{n\in\mathbb N}+ I$$, called the sum of the real numbers $$x$$ and $$y$$. The sum exists and is well-defined, i.e. it does not depend on the specific representatives $$(x_n)_{n\in\mathbb N}$$ and $$(y_n)_{n\in\mathbb N}$$ of $$x$$ and $$y$$.

Proofs: 1

Definitions: 1 2 3
Parts: 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15 16
Propositions: 17 18 19 20 21 22 23 24 25 26

Github: ### References

#### Bibliography

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