Comment by 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