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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 $(x_n)_{n\in\mathbb N}\sim (z_n)_{n\in\mathbb N}$.

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References

Bibliography

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