deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 517bbfe..a57df58 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
\begin{equation}  P(A_1) \leq P \bigg \{ \frac{1}{l}\sum_{i=1}^{l}Q(z_1, a^*) - \int Q(z,a^*)dF(z) > \frac{\epsilon}{2} \bigg \} \xrightarrow[l \rightarrow \infty]{} 0.  \end{equation}  The RHS convergencejis convergence is  valid due to the Law Of Large Numbers ($a^*$ is fixed). For bounding the probability of $A_1$ event we did not consider that uniform one-sided convergence takes place. We just used the law of large numbers.