Theorem 1

If $x$ and $y$ are any two real numbers > 0, then there is an integer $n > 0$ such that $y < nx$.

There exist two rational numbers $p/q$ and $r/s$ such that $0 < p/q < x$ and $y < r/s$, since the open intervals ]$0, x$ and ]$y, \to$ are not empty ($\S$ I, no. 4, Proposition 4); take $n$ such that $nps > qr$, and we have $y < nx$.