Originally Posted by
tonio It must be $\displaystyle a_i>0\,\,\,\forall\,\,i$, otherwise $\displaystyle 2^2\nleq (1+(-1))\left(\frac{1}{1}+\frac{-1}{1}\right)=0$
Now, do you know the Means Inequalities?:
$\displaystyle \frac{a_1+...+a_n}{n}\,\,\geq \,\,\sqrt[n]{a_1\cdot ...\cdot a_n}\,\,\geq\,\, \frac{n}{\frac{1}{a_1}+...+\frac{1}{a_n}}$
Well, taking the two extremes in the above inequalities gives you what you want. About the proof of the inequalities (well, THE inequality, since only the left one is needed: the right one follows from this taking the inverse of the elements) you can search inside MHF and look for a solution. I sent one just yesterday, in another thread)
Tonio