In the book by Apostol "calculus volume 1" how to prove that sum of two integers is an integer?

Question

In Apostol's book we start by defining a set called the set of real numbers which satisfies the field and order axioms. Then we define the set of positive integers as being the subset of every inductive set(i.e. sets which contain 1 and contain x+1 whenever it contains x) of real numbers. From the field and order axioms and the the definition of positive integers how can one prove that the sum of two positive integers is also a positive integer? Also their product? I attempted this problem by restating it as follows: Assume x,y belong to the set of positive integers. then we may reach x+y by successive operation of adding 1 to x ( i.e x+1, (x+1)+1, ((x+1)+1)+1, etc are all positive integers and x+y =((...(x+1)+1)+1)...)+1 where the plus one(+1) operation is repeated y times. How can one prove that x+y is that number which is obtained from x by this repeated operation?