Question

Prove the statement by contraposition, if a product of two positive real numbers is greater

than 100, then at least one of the numbers is greater than 10.

Answer 1

The first step in a proof by contraposition is to assume that the conclusion of the conditional statement "if a product of two positive real numbers is greater

than 100, then at least one of the numbers is greater than 10" is false; namely, assume that $a\leq10$ and $b\leq 10.$

Substituting we find that 

$$ab\leq 10(10)=100$$

This tells us that $ab\leq 100.$This is the negation of the premise of the theorem.

Because the negation of the conclusion of the conditional statement implies that the hypothesis is false, the original conditional statement is true.

Our proof by contraposition succeeded; we have proved the theorem "if a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10".