What are some good examples of sequences which are Cauchy, but do not converge?

I want an example of such a sequence in the metric space $X = \mathbb{Q}$, with $d(x, y) = |x - y|$.  And preferably, no use of series.

Another one, same idea:
$$
a_n = \left(1+\frac{1}{n}\right)^n
$$
a sequence of rationals, but its limit $e$ is not rational.