Valid example of proof by contradiction?

I'm studying "How To Prove It" (Velleman) and I'm on exercise 3.7. The solution I gave is a "proof by contradiction" and I would like to verify that the method of proof and result are valid.

Is this a well formed proof? How explicit do I have to be? For example, is it OK for me to leave out the fact that for a, b in R, a > 0 and b > 0 implies that ab > 0? If so, then what else could I have left out?

Maybe these things will be more clear once working on more complicated proofs?