Before any practice, I was once exposed to an olympiad problem that asked to prove a certain inequality. I had not a clue on how to tackle that problem. Then I found out about the AM-GM-HM inequalities, and I figured you would always have to manipulate those in order to prove a certain inequality. But surely these inequalities must be proved as well, so how is that done? I looked it up, and found out they're all based on the simple fact that a square of a real number is never negative.
So, my question is, are all inequalities based on that simple fact, in the end?