05 - Archimedean Property of ℕ

Archimedean Property of $\mathbb{N}$

$$\mathbb{N}\text{ is not bounded above}$$

Proof

Suppose for a contradiction, that $\mathbb{N}$ is bounded above
Then $\mathbb{N}$ is non-empty ($1\in \mathbb{N}\subseteq \mathbb{Z}$) and bounded above
So by completeness axiom then $\mathbb{N}$ has a supremum
By the approximation property with $ϵ=\frac{1}{2}$ then

$$\exists n\in N\text{ such that }\sup N-\frac{1}{2}<n\le \sup N$$

As $n+1\in \mathbb{N}$ and $n+1>\sup \mathbb{N}$ and hence a contradiction