P3188
We have
\begin{equation}
\begin{split}
\mathbb{E} \left( \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} \right)
& = \frac{1}{N} \sum_{n = 1}^N \mathbb{E} (I_{\{ X_n \leq x \}}) \\
& = \frac{1}{N} N \mathbb{E} (I_{\{ X_1 \leq x \}})
= \mathbb{P}(X_1 \leq x)
\end{split}
\end{equation}
$\square$
