<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
$$0\leq (a+b+c)^2=a^2+b^2+c^2+2(ab+ac+bc)$$
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