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> Let $\alpha\_1, ..., \alpha\_n \geq 0$, and $p\_1,...,p\_n \geq 0$ such
> that $\sum\_{i=1}^{n} p\_i^{-1} =1$. Show that $\alpha\_1...\alpha\_n \leq p\_{1}^{-1}\alpha\_1^{p\_1}+...+p\_{n}^{-1}\alpha\_n^{p\_n}$.
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It is Young's inequality for n numbers. I know the proof that uses AM-GM, but in this case, I can't use it.

My attempt: I tried to solve it by induction, I suppose that it was true for $n-1$ numbers and summed all the possibilities to obtain $\sum\_{k=1}^{n} \prod\_{i \in ([k]-\{k\})} \alpha\_i \leq (n-1)(p\_{1}^{-1}\alpha\_1^{p\_1}+...+p\_{n}^{-1}\alpha\_n^{p\_n})$, but don't seem to be that the left right hand is greater or equal to $\alpha\_1...\alpha\_n$.