(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# How To Prove it Inequality Proof

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