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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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