I have untouched pure math for 15 years. Real number theory is foundamental for people to do modern math. Try to understand the concept first. I can say that 50% of math undergraduates can not understand it complately even after they have graduated. You have to change the way you are thinking about.

Maybe I was missing some thing.
Sequence an is cauchy if |an -am| < e for all n, m > some N.
A real number is a class of rational sequences. In the new domain of real number, a rational number A is now (or equivalent to) a constant sequence {A, A, A, ..., A, A, ....} which can be denoted by A (now A is a notation for the rational number A), and it is always a cauchy sequence. Abstract? What does it mean by "embedding" in algebraic structure?

Usually, there are 8-10 basic theorems. You can start any one to prove all others.

Math students should learn it. However, other students do not need to learn it.

That book covers every aspect and all kind of definitions (they are all equivalent).

Natural number is created by God, but all others by human being, especially, by mathematician.