Question: Prove Theorem II. Use the method of contradiction: If the result is false, then there exists a real number x that is an upper bound of N. Use Theorem I and the definition of supremum to obtain a contradiction.
Theorem I: The supremum property implies the axiom of completeness
Theorem II: The Archimedean property. The set N of the natural numbers is not bounded above (i.e., for any x ∈ R, there exists a natural number n such that n>x).