The statement goes as follows:

$ \forall a, b > 0, a + b \geq 2\sqrt{ab} $

Which I am interpreting as "for all a and b greater than $0$, $a+b \geq 2\sqrt{ab}$".

I am now trying to reason through the logic.

So:

$a + b \geq 2\sqrt{ab}$

$ \implies a^2 + 2ab + b^2 \geq 4ab $

$ \implies a^2 + b^2 \geq 2ab $

Should I create a side proof proving that for all $ x,y > 0$, $x^2+y^2 > 2xy$. Thus, by this side proof, this is the case? 

But how would I even prove this side proof? 

Thanks in advance! 

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I feel like my proof the way I would like to present it, is way too circular.