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

[Testify]

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I originally made this thread for something else, but I have another problem that I need help with.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

 

[Mark44]

For b, you're missing the inequality in the triangle inequality.

 

[Testify]

new problem bump

 

[Tinyboss]

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 $$a<\frac1a$$. 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.