(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.
Clearly, \(\lim_{n\to\infty } x_n-x_n =0\).
Since \(\lim_{n\to\infty } y_n-x_n =0\Leftrightarrow \lim_{n\to\infty } x_n-y_n =0\).
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 \((x_n)_{n\in\mathbb N}\sim (z_n)_{n\in\mathbb N}\).
