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\begin{equation}  \inf_{a \in \Lambda(\beta_k)} \int Q(z,a)dF(z) - \inf_{a \in \Lambda(\beta_k)} \frac{1}{l} \sum_{i=1}^{l} Q(z_i,a) > \frac{\epsilon}{2}  \end{equation}  and as $T$ the event $\cup_{k=1}^nT_k$ $\cup_{k=1}^nT_k$. Then by virtue of (\ref{eq:cons_in_prob})   \begin{equation}  P(T) = P(\cup_{k=1}^n T_k) \leq \sum_{k=1}^n P(T_k) \xrightarrow[l \rightarrow \infty]{} 0  \end{equation}