# Proof

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.

Github: ### References

#### Bibliography

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