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If K is a compact set and (xn)n is a sequence in K, then (xn)n is a Cauchy sequence. a. False, and here is a counter example: K = [0, 2], (xn)n = (0, 3/2, 0, 4/3, 0, 5/4 ...). b. True, because any sequence in a compact set must converge. c. True, because Cauchy sequences are bounded and so are compact sets. d. False, and here is a counter example: K = (0, 1), (Xn)n = (1/2, 1/3, 1/2, 1/3,...).

If K is a compact set and (xn)n is a sequence in K, then (xn)n is a Cauchy sequence. a. False, and here is a counter example: K = [0, 2], (xn)n = (0, 3/2, 0, 4/3, 0, 5/4 ...). b. True, because any sequence in a compact set must converge. c. True, because Cauchy sequences are bounded and so are compact sets. d. False, and here is a counter example: K = (0, 1), (Xn)n = (1/2, 1/3, 1/2, 1/3,...).

Advanced Engineering Mathematics
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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If K is a compact set and (xn)n is a sequence in K, then (xn)n is a Cauchy sequence. a. False, and here is a counter example: K = [0, 2], (n)n = (0, 3/2, 0,4/3, 0,5/4...). b. True, because any sequence in a compact set must converge. c. True, because Cauchy sequences are bounded and so are compact sets. d. False, and here is a counter example: K = (0, 1), (Xn)n = (1/2, 1/3, 1/2, 1/3, ...).
Transcribed Image Text:If K is a compact set and (xn)n is a sequence in K, then (xn)n is a Cauchy sequence. a. False, and here is a counter example: K = [0, 2], (n)n = (0, 3/2, 0,4/3, 0,5/4...). b. True, because any sequence in a compact set must converge. c. True, because Cauchy sequences are bounded and so are compact sets. d. False, and here is a counter example: K = (0, 1), (Xn)n = (1/2, 1/3, 1/2, 1/3, ...).
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