(related to Proposition: Multiplication Of Rational Cauchy Sequences)
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.
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. ↩
