Every number is between two natural numbers

For some $y>0$, there exists a natural number $n\in \mathbb{N}$ such that $n-1\le y<n$.

Proof

By the Archimedean property, there is some $w\in \mathbb{N}$ such that $y<w$.

Let $E=\{m\in \mathbb{N}:y<m\}$ be all natural numbers of this form.

By the well-ordering principle, let $n=infE$, and n.b. $(n-1)\notin E$.

Then $n>y$ and $(n-1)\le y$.

QED.