What are the possible values of a in the inequality a < 1/a < b < 1/b?

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The inequality a < 1/a < b < 1/b establishes that if a and b are nonzero real numbers, then it can be proven that a < -1. The proof begins by assuming the inequality holds and demonstrating that a must be negative. By analyzing the implications of a < 1/a, two open intervals for a are identified, which are further refined by the conditions involving b, leading to the conclusion that a must indeed be less than -1.

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Homework Statement



I originally made this thread for something else, but I have another problem that I need help with.

Suppose a and b are nonzero real numbers. Prove that if a < 1/a < b < 1/b then a < -1.

Homework Equations



A hint was given for the problem: Assume a < 1/a < b < 1/b. Now prove that a < 0, and then use this fact to prove that a < -1.


Thanks
 
Last edited:
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For b, you're missing the inequality in the triangle inequality.
 
new problem bump
 
It's not like threads go in the landfill and pollute kindergarten playgrounds after we're done with them. No need to recycle. Make a new thread when you have a new problem.

For this problem, first think about what it means when a&lt;\frac1a. What values could a have? You should identify two possibilities (two open intervals where a could be). The second part of the inequality will let you narrow it down to one.
 

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