<s>[INST] Use the Cauchy-Schwarz Inequality to prove that $a^2+b^2+c^2 \ge ab+ac+bc $ for all positive $a,b,c$. Use the Cauchy-Schwarz Inequality to prove that $a^2+b^2+c^2 \ge ab+ac+bc $ for all positive $a,b,c$.
That's what I've tried:
Let a Cauchy-Schwarz Inequality be :
\begin{array}
(((\sqrt{a} )^2+(\sqrt{b})^2+(\sqrt{c})^2 )\left(\left(\cfrac{ab}{\sqrt{a}}\right)^2 +\left(\cfrac{ac} {\sqrt{b}}\right)^2+\left(\cfrac{bc}{\sqrt{c}}\right)^2\right)& \ge (ab+ac+bc)^2 \\
(a+b+c)(ab^2+\cfrac{a^2 c^2}{b}+b^2 c) &\ge (ab+ac+bc)^2 \\
\end{array}
However how should I now factorize the left hand side of the inequality as $(a^2+b^2+c^2) \cdot Something $ ?
I think I've made the problem harder than it needs to be .
 [/INST] If you insist on C-S, let
$$
u=(a,b,c),\quad v=(b,c,a).
$$
Then,
$$
\text{RHS}=u\cdot v\underbrace{\leq}_{\text{C-S}}||u||\times||v||=\text{LHS}.
$$
</s>