For $n \geq 2$, prove or disprove that $1 < \frac {x_1 + x_2 +... + x_n}{n} \leq 2$ ,
for all natural numbers $x_1,x_2,...,x_n$ statisfying
$${\sum \limits_{j=1}^{n}{x_j}} = {\prod \limits_{j=1}^{n}{x_j}}.$$
EDIT : As per the request of @yurnero, I hereby explicitly state that the "natural numbers" mentioned in the question mean the positive non-fraction numbers excluding $0$.
My attempt :
Let ${\sum \limits_{j=1}^{n}{x_j}}={\prod \limits_{j=1}^{n}{x_j}}=\lambda$.
Applying AM-GM on the set {$x_1,x_2,...,x_n$} :
$$\frac {\lambda}{n} \geq {\lambda}^{\frac {1}{n}}$$
$$\frac {{\lambda}^n}{n^n} \geq {\lambda}$$
$${\lambda ^{n-1}} \geq n^{n-1} . n$$
$$\lambda \geq n . n^{\frac {1}{n-1}}$$
$$\frac {\lambda}{n} \geq n^{\frac {1}{n-1}}$$
Is this approach correct? If yes, then is there any alternate way of "solving" this problem ? How to solve for the other bound/limit ?