1. The problem statement, all variables and given/known data Suppose that a and b are nonzero real numbers. Prove that if a<1/a<b<1/b then a<-1. 3. The attempt at a solution So after a while I realized that I could prove that a<-1 by contradiction but first I have to prove that a<0. I figured out how to prove it but I'm not sure if my wording is convincing enough and I feel that it might be redundant at points. Anyway, here's the proof: Proof: Suppose a<1/a<b<1/b. It then follows that 1/a<1/b. Multiplying both sides of the inequality yields b<a, but this contradicts a<b. Therefore, one and only one variable a or b must be negative, and it follows that since b>a then a<0. Now suppose a[itex]\geq[/itex]-1. Plugging the value a=-1 into a<1/a yields -1<1/-1 or -1<-1, which contradicts a<1/a. Therefore a<-1. Now I feel like the wording of that proof is a mess but I'm not sure how I would reword it. Also am I being redundant in saying "one and only one variable a or b must be negative" or does that need to be there? Or am I crazy and the proof is acceptable as is?