Sec.%2011%20Ordered%20Fields

# Sec.%2011%20Ordered%20Fields - Math 350 Advanced Calculus I...

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Math 350 Homework Solutions page 13 Section 11 Ordered Fields 11.3 (continued) (e) If x 6 = 0, then x 2 > 0. Proof: Suppose x 6 = 0. We need to show x 2 > 0 . By Axiom O1 Trichotomy Law, exactly one of the relations x = 0, x > 0, or x < 0 holds. Since x 6 = 0 by assumption, it follows that either x > 0 or x < 0. We need to prove x 2 > 0 in both of these cases. Case 1. Suppose x > 0. By Axiom O4, since x > 0, when we multiply both sides of the inequality x > 0 by x , we preserve the inequality, so we obtain x 2 = x · x > x · 0 . By Theorem 11.1(b), x · 0 = 0, so this implies x 2 > 0, as required. Case 2.
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Sec.%2011%20Ordered%20Fields - Math 350 Advanced Calculus I...

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