How to prove that $n^{-2}[x+g(x)+g\circ g(x)+\cdots +g^{\circ n}(x)]$ converges when $n\to\infty$
Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$