Describe the Dedekind cut corresponding to 27. Determine which part of the cut the following numbers lie in: 7, 6, 6.14, 3TT. Give an example of a sequence that converges to √11, and that is not monotone. Explain why f(x) = |x − 4| is continuous at 4. (Thus, f(x) = x - 4 if x > 4 and 4 - x if x ≤ 4.) Are any open subsets of R sequentially compact? Explain why or why not.
Describe the Dedekind cut corresponding to 27. Determine which part of the cut the following numbers lie in: 7, 6, 6.14, 3TT. Give an example of a sequence that converges to √11, and that is not monotone. Explain why f(x) = |x − 4| is continuous at 4. (Thus, f(x) = x - 4 if x > 4 and 4 - x if x ≤ 4.) Are any open subsets of R sequentially compact? Explain why or why not.
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:Examples. Justify your answers briefly.
(a) Describe the Dedekind cut corresponding to 27. Determine which part
of the cut the following numbers lie in: 7, 6, 6.14, 3T.
(b) Give an example of a sequence that converges to √11, and that is not
monotone.
(c) Explain why f(x) = x - 4| is continuous at 4.
(Thus, f(x) = x - 4 if x > 4 and 4 - x if x ≤ 4.)
(d) Are any open subsets of R sequentially compact? Explain why or why
not.
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