How does one prove that the natural numbers are (isomorphic to) a subset of the real numbers? First of all, I take these as the definitions of these two sets:

reals:

Field (assoc, distrib, commun, 0, 1, negatives, multiplicative inverses

Ordered (consists of three sets N, {0}, and P, where P is closed under +,x)

Complete (least upper bounds of sets are again in the reals)

naturals:

Peano's postulates would suffice.

Now, I'm already aware of the natural correspondence between {0,0+1,0+1+1,...}

in the reals and the natural numbers. What my question comes down to is: how does one prove the principle of mathematical induction for {0,1,2,...}, the subset of the reals corresponding to the naturals?

All I need in answer to this question is "vague help" or "a hint in the right direction." Or, if the answer is unbelievably complicated (which I doubt) then just let me know .