[wrsh07]

You can construct sequences of rational numbers where the limit is not rational (eg it's sqrt 2)

Trivially, the sequence of numbers who are the truncated decimal expansion of root 2 (eg 1.4, 1.41. 1.414, ...) although I find this somewhat unsatisfying.

With the real numbers there are no gaps. There are no sequences of reals where the limit of that sequence is not a real number