Show $x_1,...,x_n>0\wedge x_1\cdot...\cdot x_n=1\Rightarrow \sum x_n \geq n$

Question

Induction does not work here but why not?

Case

n=1

then $x_1=1$

Inductionstep

$x_1.....x_n+1=1\rightarrow x_1...x_n=\frac{1}{x_{n+1}}\overset{IH}{\Rightarrow}x_{n+1}=1$

This must be wrong becasuse for example

$2+\frac{1}{2}>2$ and also $1,\frac{1}{2}>0$ and $2\frac 1 2=1$ but neither is $1$. But the induction says that $x_{n+1}=1$ so the induction must be false!

I also want to know how I can solve the problem

Answer 1

Hint: Use inequality between arithmetic and geometric mean $$ {x_1+x_2+...+x_n\over n}\geq \sqrt[n]{x_1x_2...x_n}$$

Answer 2

Use the AM-GM inequalitiy $$\frac{x_1+x_2+...+x_n}{n}\geq\sqrt[n]{x_1\cdot x_2\cdot x_3\cdot …\cdot x_n}=1$$