<s>[INST] How can I prove the inequality $\frac{1}{x} +  \frac{1}{y} + \frac{1}{z} \geq \frac{9}{x+y+z}$? For $x > 0$, $y > 0$, $z > 0$, prove:
$$\frac{1}{x} +  \frac{1}{y} + \frac{1}{z} \geq \frac{9}{x+y+z} .$$
I can see that this is true, I also checked it with a few numbers. But I guess that is not enough to prove it. So how can I prove it?
(You don't need to show the whole proof, I think just a hint will be enough. I simply don't know how to start.) 
 [/INST] Since $w+\frac{1}{w} \geq 2$ for all $w>0$, we have
$$
\begin{split}
(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) 
&= 3 + \frac{x}{y}+\frac{y}{x} + \frac{x}{z}+ \frac{z}{x} +\frac{y}{z}+\frac{z}{y} \\&\geq 3+2+2+2 = 9
\end{split}
$$
</s>