Archimedean Property in Number Theory

Archimedean Property

Archimedean Property. Let $a, b \in \mathbb{N}$. Then $\exists n \in \mathbb{N}$ such that $na > b$.

해석학에서 언급하는 아르키메데스의 성질의 정수론 버전이며, $a, b$를 자연수로 가져온 것 말고 차이는 없다. 

Proof. Suppose that $\forall n \in \mathbb{N}, na \leq b$. Let $S := \{b - na \, | \, n \in \mathbb{N}\}$. Since $b - na > 0$, $\emptyset \neq S \subseteq \mathbb{N} \cup \{0\}$. Then by Well-Ordering Principle, $\exists m \in \mathbb{N}$ such that $b - ma \leq b - na, \forall n \in \mathbb{N}$. Then $n \leq m, \forall n \in \mathbb{N} \bigotimes$. $\blacksquare$