(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If [tex]x_{1} x_{2} \cdots x_{n}=1[/tex] (1)

show that

[tex]x_{1}+x_{2}+\cdots+x_{n} \geq n[/tex] (2)

3. The attempt at a solution

I attempted as follows. I started with

[tex]x_{1} + \frac{1}{x_{1}} \geq 2[/tex] , which is an inequality I already know how to prove.

Then using Eq.(1) I get

[tex]x_{1} + x_{2} x_{3} \cdots x_{n} \geq 2[/tex]

Continuing from this point , for example started from another point [tex]x_{2}[/tex] and repeating the procedure for all [tex]n[/tex] , I get no where. I cannot think of another path to take.

If i try to do it by induction, I cannot assume that the equation holds for [tex]n[/tex] numbers , and try to prove for [tex]n+1[/tex] numbers, as by including [tex]x_{n+1}[/tex], Eq.(1) and Eq.(2) need not hold anymore but

[tex]x_{1} x_{2} \cdots x_{n} x_{n+1}=1[/tex]

[tex]x_{1}+x_{2}+\cdots+x_{n} +x_{n+1}\geq n +1[/tex]

Edit:

Assuming all x's are nonnegative

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# Homework Help: Proving inequality by induction,given a condition

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