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Using (\ref{eq:cons_AtakesPlace}) and (\ref{eq:cons_eventA}) the inequalities   \begin{equation}  \label{eq:cons_AleqTk}  \begin{split}  \inf_{a \in \Lambda(\beta_k)} \int Q(z,a)dF(z) + \frac{\epsilon}{2} &  > \int Q(z,a^*)dF(z) > \frac{1}{l}\sum_{i=1}^{l}Q(z_i. a^*) + \epsilon &  > \inf_{a \in \Lambda(\beta_k)} \frac{1}{l} \sum_{i=1}^{l} Q(z_i, a) + \epsilon \end{split}  \end{equation}  hold and implie that event $T_k$ aldo takes place. This means that  \begin{equation}