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Heine-Borel Theorem. a. State the Heine-Borel Theorem b. Assume A ⊆ R and B ⊆ R...

Heine-Borel Theorem.

a. State the Heine-Borel Theorem

b. Assume A ⊆ R and B ⊆ R . Prove using only the definition of an open set that if A and B are open sets then A∩ B is an open set.

c. Assume C ⊆ R and D ⊆ R . Prove that if C and D are compact then C ∪ D is compact. There are two methods: Using the definition of compact or a proof that uses parts a. and b. above.

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