The reals are Archimedean

Statement

The ordered field $\mathbb{R}$ of real numbers is an Archimedean ordered field. That is, for all real numbers $x$ and $y$ with $y > 0$, there is a natural number $n$ with $n y > x$.

Remarks

The proof uses the completeness axiom. If $\mathbb{N}$ were bounded above, let $b = \sup \mathbb{N}$. Then $b - 1$ is not an upper bound, so $n > b - 1$ for some $n$, and $n + 1 > b$, a contradiction. Thus each complete ordered field is Archimedean.

Explanations

L1 human

However small your step and however far your goal on the real line, finitely many steps get you past the goal. There are no infinitely large real numbers.