<s>[INST] Show that inequality holds How would you show that the following inequality holds? Could you please write your reasoning by solving this problem too?
$a^2 + b^2 + c^2 \ge ab + bc + ca$  for all positive integers a, b, c
I have tried:
$a^2 + b^2 + c^2 - ab - ab + ab \ge bc + ca $
$(a^2 -2ab + b^2) + c^2 + ab \ge bc + ca $
$(a-b)^2 + c^2 + ab - bc - ca \ge 0 $
$(a-b)^2 + c(c - a) - b (c - a) \ge 0 $
$(a-b)^2 + (c - a)(c - b) \ge 0 $
Well, $(a-b)^2$ is positive. But how do we know if $(c-a)(c-b)$ is positive as well?
PS: The problem is from the book "How to think like a mathematician" by Kevin Housten.
 [/INST] Hint: Consider $(a-b)^2+(b-c)^2+(c-a)^2$. 
Remark: Using the above hint, you can show that the desired inequality holds for all real numbers $a$, $b$, and $c$.
</s>