Proving inequality by induction,given a condition

[realanony87]

Homework Statement

If $$x_{1} x_{2} \cdots x_{n}=1$$ (1)
show that
$$x_{1}+x_{2}+\cdots+x_{n} \geq n$$ (2)

The Attempt at a Solution

I attempted as follows. I started with

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

Then using Eq.(1) I get
$$x_{1} + x_{2} x_{3} \cdots x_{n} \geq 2$$

Continuing from this point , for example started from another point $$x_{2}$$ and repeating the procedure for all $$n$$ , 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 $$n$$ numbers , and try to prove for $$n+1$$ numbers, as by including $$x_{n+1}$$, Eq.(1) and Eq.(2) need not hold anymore but
$$x_{1} x_{2} \cdots x_{n} x_{n+1}=1$$
$$x_{1}+x_{2}+\cdots+x_{n} +x_{n+1}\geq n +1$$

Edit:
Assuming all x's are nonnegative

 

[morphism]

I'm assuming all the x_i are nonnegative real numbers.

Do you really want to do this by induction? It follows pretty easily from the AM-GM inequality.

 

[Dick]

Alternatively, you can use a Lagrange multiplier to extremize the sum of the xi's subject to the constraint that their product is 1.

 

[realanony87]

Well I could use AM-GM , or the method by langrange multiplier but I am interested in how to apply the principle of induction itself when such a condition is given, or if its possible at all .

 

[Dick]

I don't really see how to get at this with induction.