So I'm reading the first page of Rudin. This is right after proving that there is no rational solution to x^2=2. How does this show that the rational number system has gaps? All it shows is that A has no upper bound and B has no lower bound. Is it really necessary to have two sets A&B perfectly glued together? It seems to show that the real number line has gaps(if unfilled), but not the rational number line. Also, proving there is no rational number x such that x^2=2 is different from saying that there exists a number x such that I can take the square of it and get 2.