Suppose for the sake of contradiction the statement, For every positive $x \in\mathbb{Q}$, there is a positive $y \in \mathbb{Q}$ for which $y < x$, is false. Then the negation of this statement is true.
There is a positive $x \in \mathbb{Q}$, for every positive $y\in\mathbb{Q}$ for which $y\geq x$. Since $x$ is a positive rational number, it follows that $\frac{x}{2}$ is also a positive rational number less than $x$. Now, because $y$ ranges over all positive rational numbers, we let $y=\frac{x}{2}$. So, $\frac{x}{2} \geq x$. But this is a contradiction because $\frac{x}{2} <x$. Therefore, our assumption is false and the statement: For every positive $x \in\mathbb{Q}$, there is a positive $y \in \mathbb{Q}$ for which $y < x$ is true.