Real numbers as elements of the unique...

"One of the early objectives of almost any university mathematics course is to teach people to stop thinking of the real numbers as infinite decimals and to regard them instead as elements of the unique complete ordered field, which can be shown to exist by means of Dedekind cuts, or Cauchy sequences of rationals."

http://www.dpmms.cam.ac.uk/~wtg10/decimals.html

Could someone explain this to me using laymans terms? I'm doing basic calculus and probability atm, probably the stuff you do in the last years of school in the U.S. Maybe there are other degree level math that I could learn in my spare time that'd be abit of a leg up? If you know what I mean.

I took several university mathematics courses, and none of them mentioned that shit.  They must save that for 2nd or 3rd year courses.  Therefore the assertion is bullshit.