(related to Proposition: Rational Cauchy Sequence Members Are Bounded)

We have to show that the rational members \(a_n\) of any given rational Cauchy sequence \((a_n)_{n\in\mathbb N}\) are bounded, i.e. there exists a positive constant \(c\in\mathbb Q\), such that \(|a_n|\le c\) for all \(n\in\mathbb N\).

By definition of rational Cauchy sequences, for any rational number \(\epsilon > 0\) we have \(|a_i-a_j| < \epsilon\) for all \(i,j\ge N(\epsilon)\). In particular, if we choose \(\epsilon=1\), there is an index \(N\in\mathbb N\) for which the absolute value of the difference \(|a_n-a_m|\) becomes smaller than \(1\) for all \(m,n > N\).

Now, we consider all \(a_m\) with the index \(m=N+1\). Using the triangle property of the absolute value, we get

\[|a_n|=|a_n-a_m+a_m|\le |a_n-a_m|+|a_m| < 1+ |a_m|\quad\quad\text{for all }n > N.\] Set \(c:=\max(|a_1|,\ldots,|a_{N}|,1+|a_m|)\). By construction, the constant \(c\) gives us the required result
\[|a_n|\le c\quad\quad\text{for all }n\in\mathbb N.\]

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983