CZ-SK-MO 1996 Problem 1

A sequence $(G_n)_{n=0}^{\infty}$ satisfies $G(0) = 0$ and $G(n) = n-G(G(n-1))$ for each $n \in N$ . Show that (a) $G(k) \ge G(k -1)$ for every $k \in N$ ; (b) there is no integer $k$ for which $G(k -1) = G(k) = G(k +1)$ .