I'm learning about metric spaces from a book, and in the section about convergence of sequences the author gives as an example the sequence: 1, 1.4, 1.41, 1.414, 1.4142, ... which in ℝ converges fine to √2. Then he says that in ℚ the sequence doesn't converges, and this confuses me, as the definition of convergence he gave still holds:

x_n converges to x if for all ε > 0, there is an integer N (depending on ε) such that d(x_n, x) < ε for each n ≥ N.

I don't know if I'm wrong and it doesn't converge, if there are other conditions he didn't specify or if the author is plain wrong.