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Consider the following proposed proof of the triangle inequality (different from the proof in the textbook). Let x and y be arbitrary real numbers. The conclusion is trivial when |x + y = 0, so suppose that |x + y > 0. 1. (x + y)² = x² + 2x|ly| + y² by algebra and Proposition 1.3.1(iv). 2. x² + 2xy + y² ≥ x² + 2xy + y² = (x + y)² by Proposition 1.3.1 parts (iii) and (vi); and algebra. 3. Combining the previous two steps with Proposition 1.3.1(iv) shows that (x + y)² ≥ |x + y ². 4. The contrapositive of Proposition 1.1.8(vi) then implies the desired conclusion that |x + y ≥ x + y). Select the option that best describes this proposed proof. Step 4 is faulty. Step 3 is faulty. Step 2 is faulty. The proof is valid. Step 1 is faulty.

Consider the following proposed proof of the triangle inequality (different from the proof in the textbook). Let x and y be arbitrary real numbers. The conclusion is trivial when |x + y = 0, so suppose that |x + y > 0. 1. (x + y)² = x² + 2x|ly| + y² by algebra and Proposition 1.3.1(iv). 2. x² + 2xy + y² ≥ x² + 2xy + y² = (x + y)² by Proposition 1.3.1 parts (iii) and (vi); and algebra. 3. Combining the previous two steps with Proposition 1.3.1(iv) shows that (x + y)² ≥ |x + y ². 4. The contrapositive of Proposition 1.1.8(vi) then implies the desired conclusion that |x + y ≥ x + y). Select the option that best describes this proposed proof. Step 4 is faulty. Step 3 is faulty. Step 2 is faulty. The proof is valid. Step 1 is faulty.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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Consider the following proposed proof of the triangle inequality (different from the proof in the textbook). Let x and y be arbitrary real numbers. The conclusion is trivial when |x + y = 0, so suppose that |x + y > 0. 1. (x + y)² = x² + 2x|ly| + y² by algebra and Proposition 1.3.1(iv). 2. x² + 2xy + y² ≥ x² + 2xy + y² = (x + y)² by Proposition 1.3.1 parts (iii) and (vi); and algebra. 3. Combining the previous two steps with Proposition 1.3.1(iv) shows that (x + y)² ≥ |x + y ². 4. The contrapositive of Proposition 1.1.8(vi) then implies the desired conclusion that |x + y ≥ x + y). Select the option that best describes this proposed proof. Step 4 is faulty. Step 3 is faulty. Step 2 is faulty. The proof is valid. Step 1 is faulty.
Transcribed Image Text:Consider the following proposed proof of the triangle inequality (different from the proof in the textbook). Let x and y be arbitrary real numbers. The conclusion is trivial when |x + y = 0, so suppose that |x + y > 0. 1. (x + y)² = x² + 2x|ly| + y² by algebra and Proposition 1.3.1(iv). 2. x² + 2xy + y² ≥ x² + 2xy + y² = (x + y)² by Proposition 1.3.1 parts (iii) and (vi); and algebra. 3. Combining the previous two steps with Proposition 1.3.1(iv) shows that (x + y)² ≥ |x + y ². 4. The contrapositive of Proposition 1.1.8(vi) then implies the desired conclusion that |x + y ≥ x + y). Select the option that best describes this proposed proof. Step 4 is faulty. Step 3 is faulty. Step 2 is faulty. The proof is valid. Step 1 is faulty.
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