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A. Let p and r be real numbers, with p < r. Using the axioms of...

A. Let p and r be real numbers, with p < r. Using the axioms of the real number system, prove there exists a real number q so that p < q < r.

B. Let f: R→R be a polynomial function of even degree and let A={f(x)|x ∈R} be the range of f. Define f such that it has at least two terms.

1. Using the properties and definitions of the real number system, and in particular the definition of infimum, construct a formal proof showing inf(A) exists or explain why A does not have an infimum.

2. Using the properties and definitions of the real number system, and in particular the definition of supremum, construct a formal proof showing sup(A) exists or explain why A does not have a supremum.

C. Provide examples of two infinite bounded sets Band C and state the supremum and infimum of each.

1. Find the supremum and infimum of B∪C or explain why they do not exist.

2. Find the supremum and infimum of BnC or explain why they do not exist

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