deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 9ce0b41..3c1821c 100644  --- a/untitled.tex  +++ b/untitled.tex 
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\begin{equation}  \frac{1}{l} \sum_{i=1}^{l}Q(z, a^{**}) < \inf_{a \in \Lambda(c)} \frac{1}{l} \sum_{i=1}^{l}Q(z_i, a) + \frac{\epsilon}{2}  \end{equation}  is valid. Then, the following inequalities  \begin{equation}  \begin{split}  \frac{1}{l} \sum_{i=1}^{l}Q(z, a^{**}) + \frac{\epsilon}{2} & < \inf_{a \in \Lambda(c)} \frac{1}{l} \sum_{i=1}^{l} Q(z_i, a) +\epsilon \\  < \inf_{a \in \Lambda(c)} \int Q(a,z) dF(z) & < \int Q(z, a^{**})dF(z)  \begin{split}  \end{equation}  hold. Therefore,  \begin{equation}  \begin{split}  P(A_2) & \leq P\bigg\{ \int Q(z, a^{**})dF(z) - \frac{1}{l} \sum_{i=1}^{l}Q(z, a^{**}) > \frac{\epsilon}{2} \bigg\} \\  & leq P\bigg\{sup_{a \in \Lambda} \bigg ( \int Q(z, a^{**})dF(z) - \frac{1}{l} \sum_{i=1}^{l}Q(z, a^{**}) \bigg ) > \frac{\epsilon}{2} \bigg\} \xrightarrow[l \rightarrow \infty]{} 0  \end{split}  \end{equation}  due to uniform one-sided convergence of means to their mathematical expectations. Because,  \begin{equation}  A = A_1 \cup A_2 \Longrightarrow P(A) \leq P(A_1) + P(A_2)  \end{equation}  we have that  \begin{equation}  P(A) \xrightarrow[l \rightarrow \infty]{} 0.  \end{equation}  The theorem is proven. \hfill \Box