proof of a inequality

It's wrong.
Because (x1+x2+x3+...+xn)^2
= x1^2+x2^2+....xn^2 + 2(x1x2+x1x3+...+x1xn+x2x3+x2x4+x2..+ ..+x(n-1)xn)

Not only x_i . x_i+1

Actually you can use Schwartz inequality
with a_i= x_i, b_i=1
then Sum(i^2) Sum(x_i^2)>= [Sum(x_i)]^2
divide both sides by n and take root, you will get it.