91Ó°ÊÓ

For a set \(S\) of numbers, a member \(c\) of \(S\) is called the maximum of \(S\) provided that it is an upper bound for \(S .\) Prove that a set \(S\) of numbers has a maximum if and only if it is bounded above and sup \(S\) belongs to \(S\). Give an example of a set \(S\) of numbers that is nonempty and bounded above but has no maximum.

Use Cauchy's Inequality to prove that if \(a \geq 0\) and \(b \geq 0,\) then $$ \sqrt{a b} \leq \frac{1}{2}(a+b) $$

Use Cauchy's Inequality to show that for any numbers \(a\) and \(b\) and a natural number \(n\), $$ a b \leq \frac{1}{2}\left(n a^{2}+\frac{1}{n} b^{2}\right) $$

a. For real numbers \(a\) and \(b\), suppose that the number \(x\) is a solution to the equation $$ (x-a)(x-b)=0 $$ Prove that either \(x=a\) or \(x=b\). b. For a positive number \(c\), show that if \(x\) is any number such that \(x^{2}=c\), then either \(x=\sqrt{c}\) or \(x=-\sqrt{c}\). c. Let \(a, b,\) and \(c\) be real numbers such that \(a \neq 0\), and consider the quadratic equation $$ a x^{2}+b x+c=0 $$ Prove that a number \(x\) is a solution of this equation if and only if $$ (2 a x+b)^{2}=b^{2}-4 a c $$ Suppose that \(b^{2}-4 a c>0 .\) Prove that the quadratic equation has exactly two solutions, given by $$ x=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a} \quad \text { and } \quad x=\frac{-b-\sqrt{b^{2}-4 a c}}{2 a} $$ d. In part (c) now suppose that \(b^{2}-4 a c<0 .\) Prove that there is no real number that is a solution of the quadratic equation.

A real number of the form \(m / 2^{n}\), where \(m\) and \(n\) are integers, is called a dyadic rational. Prove that the set of dyadic rationals is dense in \(\mathbb{R}\).