Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers

In the metric space \((\mathbb Q,|~|)\), consider the sequence \((x_n)_{n\in\mathbb N}\) of (rational numbers) defined recursively by \[\begin{array}{ll} x_0 > 0 \in\mathbb Q\quad\text{(arbitrarily chosen)}\\ x_{n+1}:=\frac 12\left(x_n + \frac 2{x_n}\right) \end{array}\]

The sequence \((x_n)_{n\in\mathbb N}\) is a rational Cauchy sequence, however it is not convergent in \((\mathbb Q,|~|)\). In other words, its limit (if it exists), is not a rational number.1 Because this is one example of such a non-convergent Cauchy sequence in \((\mathbb Q,|~|)\), we can state more generally that, in the set of rational numbers, not all Cauchy sequences do converge.

Proofs: 1

Explanations: 1
Propositions: 2

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
  2. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984
  3. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013


  1. Please note that according to the corresponding lemma, all convergent sequences are also Cauchy sequences in any metric space and so in \((\mathbb Q,|~|)\). However the converse of the lemma is not true. In this proposition an example of a Cauchy sequence in the metric space \((\mathbb Q,|~|)\) is given, which has no limit in this metric space!