Suppose S is a nonvoid subset of R. Let f: R → R denote the distance to the set S; by definition, f(x) = inf{|xt|:te S}. Consider the following proposed proof that the distance function f is a Lipschitz function with Lipschitz constant equal to 1. 1. Suppose and y are real numbers, and t € S. By the triangle inequality, |æ – t| = \(2 2. By the definition of infimum, f(x) < x-t for every point t in S. 3. By transitivity of inequality, f(x) ≤ x - y + y-t| for every point t in S. −y) + (y - t) < 2 −3+y−t. 4. By the definition of infimum, f(x) ≤ x − y + f(y). 5. By symmetry, f(y) ≤ x − y + f(x), and the preceding two inequalities imply that |f(x) − f(y)| ≤ |x − y. Which one of the following statements best describes this proposed proof? Step 3 is faulty. Step 5 is faulty. Step 4 is faulty. The proof is valid. Step 1 is faulty. Step 2 is faulty.
Suppose S is a nonvoid subset of R. Let f: R → R denote the distance to the set S; by definition, f(x) = inf{|xt|:te S}. Consider the following proposed proof that the distance function f is a Lipschitz function with Lipschitz constant equal to 1. 1. Suppose and y are real numbers, and t € S. By the triangle inequality, |æ – t| = \(2 2. By the definition of infimum, f(x) < x-t for every point t in S. 3. By transitivity of inequality, f(x) ≤ x - y + y-t| for every point t in S. −y) + (y - t) < 2 −3+y−t. 4. By the definition of infimum, f(x) ≤ x − y + f(y). 5. By symmetry, f(y) ≤ x − y + f(x), and the preceding two inequalities imply that |f(x) − f(y)| ≤ |x − y. Which one of the following statements best describes this proposed proof? Step 3 is faulty. Step 5 is faulty. Step 4 is faulty. The proof is valid. Step 1 is faulty. Step 2 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:Suppose S is a nonvoid subset of R. Let f: R → R denote the
distance to the set S; by definition,
f(x) = inf{|xt|:te S}.
Consider the following proposed proof that the distance
function f is a Lipschitz function with Lipschitz constant equal
to 1.
1. Suppose and y are real numbers, and t € S. By the
triangle inequality,
|æ – t| = \(2
2. By the definition of infimum, f(x) < x-t for every
point t in S.
3. By transitivity of inequality, f(x) ≤ x - y + y-t| for
every point t in S.
−y) + (y - t) < 2 −3+y−t.
4. By the definition of infimum, f(x) ≤ x − y + f(y).
5. By symmetry, f(y) ≤ x − y + f(x), and the preceding
two inequalities imply that
|f(x) − f(y)| ≤ |x − y.
Which one of the following statements best describes this
proposed proof?
Step 3 is faulty.
Step 5 is faulty.
Step 4 is faulty.
The proof is valid.
Step 1 is faulty.
Step 2 is faulty.
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