Prove that for any real numbers $a_1, a_2,\ldots, a_n > 0$ $(n \in \mathbb{N}, n \geq 2)$ satisfying $a_1 a_2 \cdots a_n = 1$ we have $ a_1 + a_2 + a_3 + \cdots + a_n \geq n$.
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3The AM-GM-inequality should help. – Peter Oct 17 '16 at 13:57
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Do you want a proof by induction? Coz' AM-GM should do it... – SchrodingersCat Oct 17 '16 at 14:00
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Questions phrased in imperative style are not well received. Why should I do this thing you command me to do? – Myridium Oct 17 '16 at 14:03
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Not necessarily with induction. I haven't thought of AM-GM inequality, but I am trying it now. Thank you for your help! – Marc Moldovan Oct 17 '16 at 14:04
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@Myridium sorry for that. I will try to be more polite. – Marc Moldovan Oct 17 '16 at 14:07
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Since this can be thought of as the "base case" for AM-GM, you might be expected to prove this without AM-GM. (I stress: "might." ) Essentially, if you've already covered AM-GM in your class/book, or if this is a contest math problem, appealing to AM-GM is likely fine. If you haven't, it might be the questioner's intent for you to prove this case yourself. – Thomas Andrews Oct 17 '16 at 14:23