We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$.
Show that the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ does not take the value $k$ is non-zero; we denote this probability by $u$.
One may study separately the cases $p_0=0$ and $p_0>0$.