<s>[INST] Prove $a^2+b^2+c^2 \ge a+b+c$ if $abc=1$, and $a$, $b$, $c$ are positive real numbers 
Prove $a^2+b^2+c^2 \ge a+b+c$ if $abc=1$, and $a$, $b$, $c$ are positive real numbers

It is in the exercises of the AM-GM inequality chapter of a book, and that is why I believe it will be solved by that. Can anyone give me a proof using that or otherwise, too?
 [/INST] We can also proceed as follows:
\begin{align*}
a^2+b^2+c^2 &\ge \frac{(a+b+c)^2}{3}\\
&= (a+b+c)\,\frac{a+b+c}{3}\\
&\ge (a+b+c) \sqrt[3]{abc}\\
&=a+b+c.
\end{align*}
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