Is the proof correct? Either state that it is, or circle the first error and explain what is incorrect about it. If the proof is not correct, can it be fixed to prove the claim true? Claim: If f : [a, b] → R is continuous and not constant, then its range is a closed, bounded interval. Proof: Since f is continuous on a compact domain, its range is compact and hence closed and bounded. Since the range is bounded, it has a supremum and an infimum. Since the range is also closed, the supremum and infimum belong to the range. Therefore the range is the closed and bounded interval [m,M], where m is the infimum and M is the supremum of the range.
Is the proof correct? Either state that it is, or circle the first error and explain what is incorrect about it. If the proof is not correct, can it be fixed to prove the claim true? Claim: If f : [a, b] → R is continuous and not constant, then its range is a closed, bounded interval. Proof: Since f is continuous on a compact domain, its range is compact and hence closed and bounded. Since the range is bounded, it has a supremum and an infimum. Since the range is also closed, the supremum and infimum belong to the range. Therefore the range is the closed and bounded interval [m,M], where m is the infimum and M is the supremum of the range.
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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Is the proof correct? Either state that it is, or circle the first error and explain what is incorrect about it. If the proof is not correct, can it be fixed to prove the claim true?
Claim:
If f : [a, b] → R is continuous and not constant, then its range is a closed, bounded interval.
Proof:
Since f is continuous on a compact domain, its range is compact and hence closed and bounded.
Since the range is bounded, it has a supremum and an infimum.
Since the range is also closed, the supremum and infimum belong to the range.
Therefore the range is the closed and bounded interval [m,M], where m is the infimum and M is the supremum of the range.
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