Branching Process Extinction via PGF

In a branching process, each individual has $k$ offspring with probability $p_k$ ($k = 0, 1, 2, \ldots$). The probability generating function is $G(s) = \sum p_k s^k$. Prove that the extinction probability $q$ is the smallest non-negative root of $G(s) = s$. If offspring is Poisson($\lambda$), find the critical value and the extinction probability for $\lambda = 2$.