Answer
Negation: "There exists a greatest negative real number."
Proof: There is no greatest negative real number.
Suppose that $x$ is a negative real number such that, for all negative real numbers $y$, $x\geq y$. Then we can take the trivial statement $1\lt2$, multiply both sides by $x$ to get $x\gt2x$ (recalling that $x$ is negative and hence reverses the order of the inequality), and divide both sides by $2$ to get $\frac{1}{2}x\gt$x. But $\frac{1}{2}x$ must be a negative real number, because it is the product of a positive and a negative real number, so we have a contradiction. Therefore, our assumption must be false, and we conclude that there is no greatest negative real number.
Work Step by Step
This proof has many similarities to those given in Examples 4.6.1 and 4.6.3.
