What's special about real numbers is that, if you have a sequence of reals for which the distance between two consecutive elements approaches zero, then there exists a real number that's the limit of the starting sequence.
This isn't the case with rational numbers, eg. the sequence 1.4, 1.41, 1.414 ... (EDIT: these are increasing approximations of sqrt(2)) satisfies the hypothesis but there's no single rational number this sequence approaches.
This property is called completeness. The real numbers are a complete topological space, whereas the rationals aren't.