Question

Does N satisfy the Archimedean property? justify your answer.

Answer 1

We will use the definition of the Archimedean property that does not involve division because division is not defined in $\mathbb{N}$ : $\mathbb{N}$  has the Archimedean property if and only if for every positive $x\in \mathbb{N}$  and every $y\in \mathbb{N}$ , there is $n\in \mathbb{N}$  such that $y\leq nx$ .

Let $x\in \mathbb{N}$  be positive, and let $y\in \mathbb{N}$ . Since $x$  is positive and integer, $x\geq 1$ . Since $y$  is non-negative, $yx\geq y$ . Thus there is $n\in \mathbb{N}$  such that $nx\geq y$ .