- #1
realanony87
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Homework Statement
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)
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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