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.

You can't.
By assuming first what he needed to prove, your teacher has proven $True \implies True$ which is quite useless.
