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This implies [4.9], since convergence in probability implies subsequence a.e. convergence (Chung, 1974, Theorem 4.2.3).

Using Lemma 4.4, we next argue that on B∩F, player 1 believes that player 2 is eventually ignoring her history while best responding to $α^1$. In the following lemma, ε2 is from [4.6]. The set $At(τ)$ is the set of player 2 t-period histories such that player 2 ignores the next τ signals ...

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