Discussion of densely packed rationals around sqrt(2) In the first chapter of "Principles of Mathematical Analysis", Mr.Rudin includes a discussion on the set A containing all the positive rationals less than sqrt(2) and the set B containing all the positive rationals greater than sqrt(2). I understand the content, but I'm not sure about the motivation for the discussion :( How does it contribute to the discussion regarding the existence of certain gaps in the rational numbers? Why is showing that there is no largest element of A or smallest element in B important in this context?
The aforementioned discussion
Side note: This is the first time I post anything is stackexchange, so please feel free to criticize me on some conventions I need to follow in this community in the future; thank you :)
 A: I think Rudin includes the discussion there, first of all, to provide an example of a very typical analysis activity, the creation and manipulation of inequalities. The motivation for the discussion is the actual definition of the real number system, which many students never see. (It does appear in the Chapter 1 Appendix of the 3rd Edition of Rudin.)
As you know, the root of $2$ is irrational and real. One possible definition of this number is

a non-trivial binary partition of all rational numbers, the first class $A$ of which contains every $r>0: r^2 < 2$ and all non-positive rationals, while the second class $B$ contains every $r>0: r^2 > 2.$

Again, the root of $2$ is not rational; it is missing from the rational number system. But the text above is a way of describing this "certain gap" indirectly, using only concepts from the given rational number system.
If you should choose to make a study of one of the schemes for defining the real number system, you might encounter the question of whether there is, or is not, a largest element of the lower class $A$. It has technical importance when we go to show that you can do arithmetic with these non-trivial binary partitions of all rational numbers. For instance what does it really mean to find $\sqrt 3 - \sqrt2$ with such a definition of "numbers"?
Also important is an introduction to the notions of limit and convergence. If you are picturing a bounded-above set $A$ with no largest element, it seems you can travel indefinitely upwards (or rightwards if you prefer). But towards what destination are we traveling, with this sequence?
Edit: You may also find these pictures interesting: https://math.stackexchange.com/a/4123656/688046
