Homework 2: Infimums and Supremums

Instructions. Assignments should be stapled and written clearly and legibly.

1. Section 3.3, #3(b), (d), (f), (g), (h), 5, 8.
2. Suppose that \(A\) and \(B\) are two nonempty sets of real numbers such that \(x\leq y\) for all \(x\) in \(A\) and \(y\) in \(B\). In this problem you will prove that \(\operatorname{Sup} A\leq \inf B\).
   1. Give an upper bound for \(A\) and give a lower bound for \(B\).
   2. Explain how we know that both \(\operatorname{Sup} A\) and \(\inf B\) must exist.
   3. Prove that \(\operatorname{Sup} A\leq y\) for all \(y\in B\).
   4. Use part (c) and the definition of \(\inf B\) to prove that \(\operatorname{Sup} A\leq \inf B\).
   5. Can one say that \(\max A\leq \min B\)? Justify your answer.