Consider the sequence of partial sums $s_n = \sum_{k=1}^{n} \frac{1}{k^2}$. As a finite sum of rational numbers, the $s_n$ itsself are rational. However, since $\pi$ is transcendental, $\pi^2$ is an irrational number.
We can show that $s_n$ is Cauchy and $s_n \to \pi^2/6$, i.e. we have a cauchy sequence lying in $\mathbb{Q}$, which does not converge in $\mathbb{Q}$.
Is this a valid example, to show the incompleteness of the rational numbers?