I originally made this thread for something else, but I have another problem that I need help with.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