Why does this way of solving inequalities work? Here is what I had to prove.
Question: For positive reals $a$ and $b$ prove that $a^2+b^2 \geq 2ab$.
Here is how my teacher did it: 

First assume that it is in fact, true that $a^2+b^2 \geq 2ab$. Therefore $a^2+b^2-2ab \geq 0$ . We have $(a-b)^2$ is greater than or equal to zero which is true. Hence what was assumed originally is true.

Why does this method work?
I cannot understand how you can first assume that it is true and then prove that it is.

The way it's written is wrong, and you're right, we can't assume something and prove that it is right. 
However, instead of one way implication at each step (like $a^2+b^2\geq 2ab$ therefore $a^2+b^2 -2ab\geq 0$), we in fact have an if and only if condition. Since each statement then becomes equivalent, if you can prove any one of them you are done. And that is the case since $(a-b)^2$ is indeed greater than equal to $0$.
