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Show that the assertion of the Heine-Borel Theorem is equivalent to the completeness axiom for the real numbers. Show that the assertion of the Nested set theorem is equivalent to the completeness axiom for the real numbers

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Show that the assertion of the Heine-Borel Theorem is equivalent to the completeness axiom for the real numbers. Show that the assertion of the Nested set theorem is equivalent to the completeness axiom for the real numbers

Problem 4. Prove using the completeness property that √2 is a real number by considering the set {x∈(0,2)|x^2<2}.

Use the Completeness Axiom to prove that every nonempty set of real numbers that is bounded below has an infimum.

A) State the completeness Axiom for real numbers, prove that the infimum of the interval (3,7) equal 3.

Infimum version of the Axiom of Completeness. Assuming the Axiom of Completeness, prove that any nonempty set of real numbers A which is bounded below has a greatest lower bound. Hint. Consider the set {−a : a ∈ A}.

Problem 5. Show that the assertion of the Heine-Borel theorem is equivalent to the completeness Axiom for the real numbers.

Question. Open exercise 2.18 let (a_n) be an increasing bounded sequence.by the completeness Axiom the sequence converges to some real numbers a. show that its range {a_n| n∈N} has a supremum and that the supremum equals a

Give five consequences of the completeness axioms of real numbers.

Show that the assertion of the Heine-Borel Theorem is equivalent to the completeness axiom for the real numbers. Show that the assertion of the Nested set theorem is equivalent to the completeness axiom for the real numbers

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