Chapter 7.1, Problem 26E

Prove that if $a$ and $b$ are real numbers such that $a>b$, then there exist a rational number $m/n$ such that $a>m/n>b$. (Hint: Use Exercise 25 to obtain $n\in {\mathbb{Z}}^{+}$ such that $a-b>1/n$. Then choose $m$ to be the least integer such that $m>nb$. With these choices of $m$ and $n$, show that $(m-1)/n\le b$ and then that $a>m/n>b$.)

If $a$ and $b$ are positive real numbers, prove that there exist a positive integer $n$ such that $na>b$. This property is called Archimedean Property of the real numbers.

(Hint: If $ma\le b$ for all $m\in {\mathbb{Z}}^{+}$, then $b$ is an upper bound for the set $S=\left\{ma\right|m\in {\mathbb{Z}}^{+}\}$. Use the completeness property of $ℝ$ to arrive at a contradiction.)