$\sum _{k=1}^{\tau}{E}_{\tilde{P}}[1-{E}_{\tilde{P}}[{\sigma}_{2}({h}_{2}^{t+k})({\widehat{a}}_{2})|{h}_{1}^{{T}_{1}},B]\to 0\text{ast}\to \infty$

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 ${\widehat{\alpha}}_{1}$. In the following lemma, ε_{2} is from [4.6]. The set ${A}_{t}(\tau )$ is the set of player 2 t-period histories such that player 2 ignores the next τ signals ...