(related to Proposition: Addition of Rational Cauchy Sequences)
Let \((x_n)_{n\in\mathbb N}\) and \((y_n)_{n\in\mathbb N}\) be rational Cauchy sequences. By definition, it means that, for any arbitrarily small rational number \(\epsilon > 0\), there exist two natural numbers \(N_x(\epsilon/2)\) and \(N_y(\epsilon/2)\) such that for all \(n,m\in\mathbb N\) with \(n, m > \max(N_x(\epsilon/2),N_y(\epsilon/2))\) we have \[|x_n - x_m| < \frac\epsilon2 \quad\text{and}\quad |y_n - y_m| < \frac\epsilon2. \]
Note that, since \(x_n\) and \(y_n\) are rational numbers for all \(n\in\mathbb N\), it follows from the definition of addition of rational numbers that \((x_n - x_m)\), \((y_n - y_m)\), \((x_n + y_n)\) are also rational numbers. Therefore, the sequence \((x_n+y_n)_{n\in\mathbb N}\) is a sequence of rational numbers. Using the triangle inequality for the absolute value of rational numbers, we estimate \[|(x_n + y_n) - (x_m + y_m)| \le |x_n - x_m|+|y_n - y_m| < \frac\epsilon2 + \frac\epsilon2=\epsilon.~~~~~~~~~~~~( * )\] From \( ( * ) \) it follows that the rational sequence \((x_n + y_n)_{n\in\mathbb N}\) is a rational Cauchy sequence.
