91Ó°ÊÓ

Using just the axioms, prove the arithmetic-geometric mean inequality: $$ \sqrt{a b} \leq \frac{a+b}{2} $$ for any \(a, b \in \mathbb{R}\) with \(a>0\) and \(b>0\). (Assume, for the moment, the existence of square roots.)

$$ \text { If a set } E \text { is dense, what can you conclude about the set } \mathbb{R} \backslash E ? $$

The mathematician Leibniz based his calculus on the assumption that there were "infinitesimals," positive real numbers that are extremely small - smaller than all positive rational numbers certainly. Some calculus students also believe, apparently, in the existence of such numbers since they can imagine a number that is "just next to zero." Is there a positive real number smaller than all positive rational numbers?

Let \(A\) be a set of real numbers and let \(B=\\{x+r: x \in A\\}\) for some number \(r\). Find a relation between \(\sup A\) and \(\sup B\).

Let \(A\) and \(B\) be sets of real numbers and write \(C=A \cap B\). Find a relation among \(\sup A, \sup B\), and \(\sup C\).

A function is said to be bounded if its range is a bounded set. Give examples of functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) that are bounded and examples of such functions that are unbounded. Give an example of one that has the property that $$ \sup \\{f(x): x \in \mathbb{R}\\} $$ is finite but \(\max \\{f(x): x \in \mathbb{R}\\}\) does not exist.