I'm sorry I didn't reply to this earlier KnifeMissile. Try reading the beginning of any *serious* book on Analysis. A lot of books do introduce real numbers by the axioms, but others take the time to construct them from the rationals, for instance by means of Cauchy series.
Here's a definition of the order on real numbers: Given two real numbers x and y, say that x<=y if there is another real number z such that y-x=z*z. If there is no such z, say that x>y. Equal if their equal. Now what you would have to prove with this approach (which isn't the best, I'm just giving you something you might not find in the books), is that if x<y, then y>x.
By the way, the "least upper bound principle" is what is also called the "existence of suprema", and this too is proven in serious books. The reason I'm not giving you precise references is that I'm from Switzerland and know the German much better than the English literature.
Here's a resumee: If you want to get to the applications of real numbers quickly, you can introduce real numbers by axioms. But this is only cutting corners, and mathematicians know how to prove that the real numbers constructed by one of the standard procedures (Dedekind cuts, or Cauchy sequences) actually fulfill those axioms, and what's more, are completely characterised by them.