I have a proof by contrapositive (must be contrapositive) for the statement above, but I have been told that there is an error in my proof. Can someone verify this proof for me, and perhaps shed some light as to where this error is?
Proof, by contrapositive.
1. Instead of the original statement, we prove the contrapositive. 2. If $n^2$ is odd, then $n$ is odd. 3. Assume that $n^2$ is odd. 4. Claim $n$ is odd. 5. By assumption, $n^2 = n \cdot n$ is odd. 6. If a product is odd, then each factor is odd. 7. Therefore, $n$ is odd.