Using field axioms and order axioms prove the following theorems(explain every step by referencing basic axioms)
(i) The sets R (real numbers), P (positive numbers) and [1,infinity) are all inductive
(ii) N (set of natural numbers) is inductive. In particular, 1is a natural number
(iii) If n is a natural number, then n >= 1
(iv) (The induction principle). If M is a subset of N (set ofnatural numbers) then M = N
The following definitions are given:
A subset S of R is called inductive, if 1 is an element of S andif x + 1 is an element of S whenever x is an element of S.
The intersection of all inductive sets if called the set ofnatural numbers and is denoted by N
