I have seen in many textbooks on analysis that the Archimedean property of reals is a consequence of the completeness axiom. However I am not convinced that we need to use such a powerful axiom (as the completeness axiom) to prove a very basic property like Archimedean Property. To me it looks simple enough as follows:
1) If a and b are natural numbers (like 1, 2, 3, ..., but not 0) and there is another natural number c such that b = a + c, then we write b > a. Using this we can see that given any natural number we can find another greater than it.
2) If a & b are positive rationals then a/b is rational and we can write a/b = p/q (p, q positive integers) and by division algorithm we have a non-negative integer p = qx + r with 0 <= r < q. Then clearly we have a positive integer (x + 1) > p/q = a/b. So that field of rationals possesses the Archimedean property.
3) If a, b are positive reals then a/b is also real. Any definition of real numbers (Dedekind's or Cauchy's for example) will lead to the fact that given a real number there is a rational greater than it and a rational less than it. So we have rational c > a/b. And clearly by Archimedean Property of rationals (point 2 above) we have a positive integer 'n' greater than 'c'. Thus we finally have a/b < n or a < nb.
I am really not sure why textbooks try to prove Archimedean property via completeness axiom. If it was really the case Archimedean property is a consequence of completeness property, then how come rational possess Archimedean property but not completeness axiom.
Please clarify if there is any mistake in above reasoning or the textbooks are treating this in high handed manner.