Do the following task. a) State formally the Heine-Borel theorem in R^k b) Apply Heine-Borel theorem to show that if A is bounded, B is closed such that B ⊆ A ⊆ R^k, then B is compact. c) Give an example of set A and B in R such that A is bounded, B is closed, B ⊆ A but A is not compact.

Do the following task. a) State formally the Heine-Borel theorem in R^k b) Apply Heine-Borel theorem to show that if A is bounded, B is closed such that B ⊆ A ⊆ R^k, then B is compact. c) Give an example of set A and B in R such that A is bounded, B is closed, B ⊆ A but A is not compact.

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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Question

Do the following task.
a) State formally the Heine-Borel theorem in R^k
b) Apply Heine-Borel theorem to show that if A is bounded, B is
closed such that B ⊆ A ⊆ R^k, then B is compact.
c) Give an example of set A and B in R such that A is bounded,
B is closed, B ⊆ A but A is not compact.

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