I've tried reading Rudin a couple times, but it's a bit of a slog. There are easier analysis texts. Abbott's Understanding Analysis is good, though a bit basic. I'm currently reading Terence Tao's Analysis I, which is very good if you're in the right frame of mind for it. The first 150 pages are spent building the real numbers from scratch, starting with set theory and the Peano axioms. You successively construct the naturals, the integers, the rationals, and then the reals (as equivalence classes of Cauchy sequences of rationals). It's fun to see how the sausage is made, but I can also admit that when I was just starting out in math I might have found this book unbearably tedious.