## 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}.

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# Calculus

## 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

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