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 $n$. We define $u_n$, $u_n^{(r)}$, $U(s)$ and $U_r(s)$ as above, and $u$ is the probability that $(W_n)$ does not take the value $k$.
Show that the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ infinitely many times is zero.

We assume $m>1$. We study the Galton-Watson process with $Y_n$ the number of individuals in generation $n$ (starting from 1 individual).
Show that the probability that the sequence $(Y_n)_{n\in\mathbb{N}^*}$ takes any fixed value $k$ infinitely many times is zero.

Let $(A_n)_{n\in\mathbb{N}}$ be a sequence of events all with probability 1.
Show that $P\left(\bigcup_{n\in\mathbb{N}}\overline{A_n}\right)=0$. What can be deduced for $P\left(\bigcap_{n\in\mathbb{N}}A_n\right)$?

We assume $m>1$. Let $\alpha$ be the probability of extinction and $\beta$ be the probability that the sequence $(Y_n)$ diverges to infinity.
Show that $\alpha+\beta=1$.