1. ## Limit 3

Let $\displaystyle f$ be function for which which holds following conditions

$\displaystyle F\in C^1(R)\mbox{ and }|f'(x)|<1, \forall x\in R$
$\displaystyle f(x+1)=f(x)\ ,\forall x\in R, f \mbox{ is periodical, with } \omega=1)$

Lets define function $\displaystyle p:R\rightarrow R$ with $\displaystyle p(x)=x+f(x)$.
Prove that
$\displaystyle \displaystyle \lim\limits_{n\to\infty}\frac{{x+p(x)+p(p(x))+\dot s + \overbrace{p(p(\dots (p(x))\dots))}^{(n-1)-\text{times}}}}{{n^2}}$

exist, and doesn't depends on $\displaystyle x$.