I'm reading the paper: "Model Checking Games for Branching Time Logics" by Martin Lange and Colin Stirling - https://carrick.fmv.informatik.uni-kassel.de/~mlange/papers/jlc2000.pdf. The paper defines a model checking game for $\text{CTL}^*$. The main result of the paper is the following (page 10):
$$\textbf{Theorem 16 }\; \text{Player } \mathrm{II}\; \text{wins } \Gamma_{\text{CTL}^*}(\mathcal{T},s,\varphi_0) \text{ iff } \mathcal{T},s \models \varphi_0. $$
While I have intuition for why this is true, I don't understand the formal proof given in the paper. The problem is reduced to only proving one direction (the "if" part), and only considering formulas of the form $\varphi_0 = A\varphi$, where $\varphi$ doesn't have any path quantifiers (i.e. it is an LTL formula). This reduction is understood. Then, a winning strategy for player $\mathrm{II}$ is defined under the assumption that $\mathcal{T},s \models \varphi_0$. Then, the correctness of the strategy is shown.
This is where my main problem with the proof begins. Specifically, in the following sentence (page 11):
Since $s \models \varphi_0$ there is a $\alpha \in C$ s.t. $\sigma^{\omega}\models \alpha$. We show that player $\mathrm{II}$ is able to find this $\alpha$ and win before player $\mathrm{I}$ can win the play with winning condition (3).
The proof then proceeds with what seems to be an induction on the structure of $\alpha$.
I understand that for every configuration in the play, there is at least one formula ($\alpha$) that is satisfied by the suffix of the corresponding path in that play. But as for the second part of the above sentence, I don't understand what is the exact induction hypothesis that is being used.
If anyone is familiar with this paper/topic, I will very appreciate any guidance in understanding this proof, as I've been cracking my head over it for a while. I've also tried to find alternative ways to prove it, but without success.
