[Math] Rudin, Real Analysis, 3ed - Problem 2.1.1

2: Chapter 2 - The Real Number System
1: Section 1 - Axioms for the Real Numbers
1: Problem 1 - Show that 1 in P.

((1 is positive))

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My Show:

A. The Field Axioms: A7 there exists a real number 1 not equal to 0 and x*1 = x for all real x.

B. Axioms of Order: B4. 1 is a real number => (1 = 0) or (1 in P) or (-1 in P). It cannot be 1 = 0 by A7. By B2, whether (1 in P) or (-1 in P), 1 = 1*1 = (-1)*(-1) in P.

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