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4. (10 points) Prove that if A is not connected, then A is not an interval. 5. (10 points) Use only the definition of a compact set to prove the interval [0, o0) is not compact. 6. (10 points) Let {an} be a sequence in R with rn a. Use the definition of a compact set to prove the set {Tn :n € N}U {a} is compact. 7. (10 points) Let K be a nonempty compact subset of R. Prove every sequence in K has a convergent subsequence that converges in K.

4. (10 points) Prove that if A is not connected, then A is not an interval. 5. (10 points) Use only the definition of a compact set to prove the interval [0, o0) is not compact. 6. (10 points) Let {an} be a sequence in R with rn a. Use the definition of a compact set to prove the set {Tn :n € N}U {a} is compact. 7. (10 points) Let K be a nonempty compact subset of R. Prove every sequence in K has a convergent subsequence that converges in K.

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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4. (10 points) Prove that if A is not connected, then A is not an interval. 5. (10 points) Use only the definition of a compact set to prove the interval (0, o0) is not compact. 6. (10 points) Let {an} be a sequence in R with rn- {rn :n E N}U{a} is compact. → a. Use the definition of a compact set to prove the set 7. (10 points) Let K be a nonempty compact subset of R. Prove every sequence in K has a convergent subsequence that converges in K.
Transcribed Image Text:4. (10 points) Prove that if A is not connected, then A is not an interval. 5. (10 points) Use only the definition of a compact set to prove the interval (0, o0) is not compact. 6. (10 points) Let {an} be a sequence in R with rn- {rn :n E N}U{a} is compact. → a. Use the definition of a compact set to prove the set 7. (10 points) Let K be a nonempty compact subset of R. Prove every sequence in K has a convergent subsequence that converges in K.
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