Can you prove the Archimedean property of the rational numbers without constructing the reals and using the least upper bound property? It seems odd to have to take this roundabout approach, but I don't know any proof that avoids it.
[Math] Archimedean property of the rational numbers
analysis
Best Answer
Every positive rational number is of the form $m/n$ where $m$, $n$ are positive integers. If you add up more than $n$ copies of this, the sum is more than 1, so there you have the Archimedean property.