Chapter 7.1, Problem 25E

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.)