Proving there is no negative real number greater that all negative rational numbers How would I use the Archimedean property to prove that there is no negative real number greater that all negative rational numbers?

Let $x$ be any negative real number. Then $-x$ is positive, and by the Archimedean property there is a natural number $n$ such that $0<-x<n$.
Then $-\frac 1n$ is a negative rational number, and
$$x<-\frac 1n<0$$
Thus $x$ is not greater than all negative rational numbers. Since $x$ are arbitrary, there is no negative real number greater that all negative rational numbers.
