= Solution
Evaluation of the <probability generating function> at zero selects the constant coefficient, so
$$
\boxed{\mathbb P(X_n=0)=G_n(0)=1-\alpha^{(1-\beta^n)/(1-\beta)}\longrightarrow1-\alpha^{1/(1-\beta)}.}
$$
To identify this limit with ultimate extinction, let $E_n=\{X_n=0\}$. The absorbing-zero property of the <branching process> gives $E_n\subseteq E_{n+1}$. Ultimate extinction means that some finite generation is empty, namely $E=\bigcup_{n\ge0}E_n$. Continuity of <probability> on increasing <events> therefore gives
$$
\boxed{\mathbb P(E)=\lim_{n\to\infty}\mathbb P(E_n)=1-\alpha^{1/(1-\beta)}.}
$$
This set argument is essential: it uses absorption, rather than inferring extinction merely from a numerical limit of generation probabilities.
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