Let D denote the field of real numbers, and let D^p be any subset of D that satisfies the conditions of the definition here:
(1)closure under addition if a,b exits in D^P, then a + b exist D^P
(2)closure under multiplication if a, b exist in D^p, then ab exist in D^p
(3)law of trichotomy if a exist in D, then exactly one of the following is true: a = 0, a exist in D^p or -a exist in D^p
Prove that D^p = (0, infinity)(In other words, the usual set
of positive real numbers is the only subset of D = R that can serve as D^p).
Please help! We are stuck :(