# Proof

(related to Proposition: Definition of Real Numbers)

Let $$(x_n)_{n\in\mathbb N}$$, $$(y_n)_{n\in\mathbb N}$$ be rational Cauchy sequences. We will prove that the relation defined by

$(x_n)_{n\in\mathbb N}\sim(y_n)_{n\in\mathbb N}\quad\Longleftrightarrow\quad \lim_{n\to\infty } y_n-x_n =0\in\mathbb Q$

is an equivalence relation. In other words, the real numbers as equivalence classes $$x:=(x_n)_{n\in\mathbb N}+I:=\{(y_n)_{n\in\mathbb N},~ (y_n)_{n\in\mathbb N}\sim (x_n)_{n\in\mathbb N}\}$$ are well-defined.

### $$( i )$$ Reflexivity

Clearly, $$\lim_{n\to\infty } x_n-x_n =0$$.

### $$( ii )$$ Symmetry $$(x_n)_{n\in\mathbb N}\sim (y_n)_{n\in\mathbb N}\Leftrightarrow (y_n)_{n\in\mathbb N}\sim (x_n)_{n\in\mathbb N}$$

Since $$\lim_{n\to\infty } y_n-x_n =0\Leftrightarrow \lim_{n\to\infty } x_n-y_n =0$$.

### $$( iii )$$ Transitivity

By hypothesis, the rational Cauchy sequences $$(i_n)_{n\in\mathbb N}:=(y_n - x_n)_{n\in\mathbb N}$$ and $$(j_n)_{n\in\mathbb N}:=(z_n - y_n)_{n\in\mathbb N}$$ are convergent to $$0$$. Because $$z_n-x_n=j_n+i_n$$ and because $$\lim_{n\to\infty} j_n+i_n=0$$, it follows $$(x_n)_{n\in\mathbb N}\sim (z_n)_{n\in\mathbb N}$$.

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

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