# Proof

It had been shown already that all rational Cauchy sequences are bounded. In particular, for any two given rational Cauchy sequences $$(x_n)_{n\in\mathbb N}$$ and $$(y_n)_{n\in\mathbb N}$$, we can find rational constants $$c_x > 0,c_y > 0\in\mathbb Q$$ with $|x_n|\le c_x\quad\text{respectively}\quad|y_n|\le c_y\quad\quad\text{for all }n\in\mathbb N.$

By definition of rational Cauchy sequences, for any arbitrarily small rational number $$\epsilon > 0$$, we can find two natural numbers $$N_x(\epsilon/(2c_x))$$ and $$N_y(\epsilon/(2c_y))$$ such that for all $$n,m\in\mathbb N$$ with $$n, m > N(\epsilon):=\max(N_x(\epsilon/(2c_x)),N_y(\epsilon/(2c_y)))$$ we have1 $|x_n - x_m| < \frac\epsilon{2c_x} \quad\text{and}\quad |y_n - y_m| < \frac\epsilon{2c_y}\quad\quad( * ).$

With this estimation, and using the following mathematical definitions and concepts: * rational $$0$$ is neutral with respect to the addition of rational numbers, * definition of adding rational numbers, * distributivity law for rational numbers, * properties of the absolute value for rational numbers, and * definition of multiplying integers:

$\begin{array}{ccll} |x_n \cdot b_n - x_m \cdot b_m|&=& |x_n \cdot y_n + 0 - x_m \cdot y_m|&\text{because }0\text{ is neutral with respect to the addition of rational numbers}\\ &=& |x_n \cdot y_n - x_n\cdot y_m + x_n\cdot y_m - x_m \cdot y_m|&\text{adding (inverse) rational numbers results in }0\\ &=&|x_n\cdot(y_n-y_m)+y_m\cdot(x_n-x_m)|&\text{by distributivity law for rational numbers}\\ &\le&|x_n\cdot(y_n-y_m)|+|y_m\cdot(x_n-x_m)|&\text{by triangle inequality}\\ &=&|x_n|\cdot|y_n-y_m|+|y_m|\cdot|x_n-x_m|\\ &\le& c_x\cdot\frac\epsilon{2c_x} + c_y\cdot\frac\epsilon{2c_y}&\text{by estimation above }( * )\\ &=& \frac\epsilon{2} + \frac\epsilon{2}&\text{by multiplication of rational numbers}\\ &=&\epsilon&\text{by addition of rational numbers}\\ \end{array}$

As this estimation is valid for all $$n, m > N(\epsilon)$$, it follows that the sequence of such products $$(x_n \cdot y_n)_{n\in\mathbb N}$$ 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

#### Footnotes

1. Note that both, $$c_x > 0, c_y > 0$$ by construction, so we do not have to be afraid of dividing by zero in the above reasoning.