Consider the following proposed proof that a continuous function f: [0, 1] → R is bounded (special case of Lemma 3.3.1). 1. Since the definition of continuity is universally quantified on ɛ, the definition can be applied with & equal to 1. 2. The definition of continuity provides a positive & such that if a and c are points of the interval [0, 1] for which | - c < 8, then f(x) = f(c) < 1. 3. By the Archimedean property, there is a natural number n such that 1/n < 8. 4. Let M denote the maximum of the numbers |f(0/n), f(1/n)\, .... f(n/n). 5. Every x in the interval [0, 1] differs from one of the numbers 0, 1/n, 2/n,..., n/n by less than 8, so steps (2) and (4) and the triangle inequality imply that f(x)| ≤ M + 1. Which one of the following statements best describes this proposed proof? Step 3 is faulty. Step 2 is faulty. Step 4 is faulty. The proof is valid. Step 5 is faulty. Step 1 is faulty.
Consider the following proposed proof that a continuous function f: [0, 1] → R is bounded (special case of Lemma 3.3.1). 1. Since the definition of continuity is universally quantified on ɛ, the definition can be applied with & equal to 1. 2. The definition of continuity provides a positive & such that if a and c are points of the interval [0, 1] for which | - c < 8, then f(x) = f(c) < 1. 3. By the Archimedean property, there is a natural number n such that 1/n < 8. 4. Let M denote the maximum of the numbers |f(0/n), f(1/n)\, .... f(n/n). 5. Every x in the interval [0, 1] differs from one of the numbers 0, 1/n, 2/n,..., n/n by less than 8, so steps (2) and (4) and the triangle inequality imply that f(x)| ≤ M + 1. Which one of the following statements best describes this proposed proof? Step 3 is faulty. Step 2 is faulty. Step 4 is faulty. The proof is valid. Step 5 is faulty. 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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![Consider the following proposed proof that a continuous function
f: [0, 1] → R is bounded (special case of Lemma 3.3.1).
1. Since the definition of continuity is universally quantified on ɛ, the
definition can be applied with & equal to 1.
2. The definition of continuity provides a positive & such that if a
and care points of the interval [0, 1] for which x − c < 8, then
f(x) = f(c) < 1.
3. By the Archimedean property, there is a natural number n such
that 1/n < 8.
4. Let M denote the maximum of the numbers |ƒ(0/n)|, |f(1/n)|,
.... f(n/n).
5. Every x in the interval [0, 1] differs from one of the numbers 0,
1/n, 2/n,..., n/n by less than 8, so steps (2) and (4) and the
triangle inequality imply that f(x)| ≤ M + 1.
Which one of the following statements best describes this proposed
proof?
Step 3 is faulty.
Step 2 is faulty.
Step 4 is faulty.
The proof is valid.
Step 5 is faulty.
Step 1 is faulty.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b19493d-2e0b-4edb-977e-29215d3bb0dd%2F6326ebed-0366-419e-b901-492b88673e2e%2Fvrh7i1x_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following proposed proof that a continuous function
f: [0, 1] → R is bounded (special case of Lemma 3.3.1).
1. Since the definition of continuity is universally quantified on ɛ, the
definition can be applied with & equal to 1.
2. The definition of continuity provides a positive & such that if a
and care points of the interval [0, 1] for which x − c < 8, then
f(x) = f(c) < 1.
3. By the Archimedean property, there is a natural number n such
that 1/n < 8.
4. Let M denote the maximum of the numbers |ƒ(0/n)|, |f(1/n)|,
.... f(n/n).
5. Every x in the interval [0, 1] differs from one of the numbers 0,
1/n, 2/n,..., n/n by less than 8, so steps (2) and (4) and the
triangle inequality imply that f(x)| ≤ M + 1.
Which one of the following statements best describes this proposed
proof?
Step 3 is faulty.
Step 2 is faulty.
Step 4 is faulty.
The proof is valid.
Step 5 is faulty.
Step 1 is faulty.
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