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 not actually. You should rather start with $(a-b)^2 \geq 0$ and the show that $a^2+b^2 \geq 2ab$, not the other way round. 
