Question

Prove by contradiction that a real number that is greater than every negative real number cannot be negative.

Answer 1

Answer:

We wish to prove by contradiction that a real number that is greater than every negative real number cannot be negative.

Proving something by contradition means it will be supposed that what we are trying to prove is false and then get a contradiction with the hypothesis. Let's do it!

Suppose there is a real negative number $a$ that is greater than every negative number. By definition of being negative, we have a<0. Dividing both side by $2$, we get $\frac{a}{2}<0$. That is, $\frac{a}{2}$ is also a real negative number.

But $\frac{a}{2}>a$ (once they are negative). Then we found a negative real number greater than $a$, what is a contradiction, once we supposed $a$ is greater then every negative real number.

Therefore, a real number that is greater than every negative real number cannot be negative.