Simple AM-GM inequality Let $a,b,c$ positive real numbers such that $a+b+c=3$, using only AM-GM inequalities show that
$$
a+b+c \geq ab+bc+ca
$$
I was able to prove that
$$
\begin{align}
a^2+b^2+c^2 &=\frac{a^2+b^2}{2}+\frac{b^2+c^2}{2}+\frac{a^2+c^2}{2} \geq \\
&\ge  \frac{2\sqrt{a^2b^2}}{2}+\frac{2\sqrt{b^2c^2}}{2}+\frac{2\sqrt{a^2c^2}}{2} \\
&= ab+bc+ca
\end{align}
$$
but now I am stuck. I don't know how to use the fact that $a+b+c=3$ to prove the inequality. Anybody can give me a hint?

Reformulate first $a+b+c=S$ and $a=M+d \qquad b=M-d $ then 
$$S  \ge M^2-d^2 + (S-2M)2M$$
reorganize
$$ 3M^2-2SM+S  +d^2  \ge 0 $$
Now make use of the given definition that $S=3$. We get
$$3(M^2-2M+1)  +d^2  \ge 0 $$
$$3(M-1)^2 +d^2  \ge 0 $$
which is always true.   
Well, this focuses "when and how" it makes sense to introduce the condition that $S=3$. Unfortunately the step with the AM-GM-inequality is lost. But maybe you can combine your steps with this derivations? 
