Is this correct method to prove that $a^2 + b^2 + c^2 ≥ ab + bc + ac$, when $a,b,c \geq 0$? Can I prove it like this: Let's say that $a=b=c$ so we get "If $a \geq 0$ then $3a^2 ≥ 3a^2$" Now I take the negation of that statement and get "If $a \geq 0$ then $3a^2 < 3a^2$" The anti-thesis is obviously wrong which makes the original thesis right? Is this a correct way to do this, if not can you give some tips for what to do. I am not looking for complete solution as these are my homework and I really need to practice. 

Hint: We have equality when $a=b=c$. So perhaps express the difference in terms of $a-b$, $b-c$, and $c-a$.
Remark: One cannot expect to be able to prove that something holds for all $a,b,c$ by looking only at a special case. And one cannot expect to prove a mathematical result by logical manipulation: the informal mathematical idea comes first, with logic playing a supporting role. 
