# Rationals are Archimedean.
The rational numbers itself is an Archimedean field: For any rational $r$ and positive rational $p > 0$, there exists an integer $N$ such that $Np > r$.
$\blacktriangleright$ Indeed, suppose $r = \frac{a}{M}$ and $p = \frac{b}{M} > 0$ for some common denominator $M > 0$. If $r \le 0$, take $N = 1$. If $r > 0$, then $a > 0$, and note $$
(a+1) M p = (a+1) M \frac{b}{M}= (a+1) b > ab \ge a \ge r.
$$So take $N = (a+1) M$. $\blacksquare$
B / 7 2024