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
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! ↩