The random indicator $I_{\{ X_n \leq x \}}$ has expectation
\begin{equation}
\mathbb{E}(I_{\{ X_n \leq x \}}) = 0 \cdot \mathbb{P}(X > x) + 1 \cdot \mathbb{P}(X_n \leq x) = \mathbb{P}(X_1 \leq x)
\end{equation}
The claim is thus a consequence of
R2410: I.I.D. real weak law of large numbers under finite second absolute moments. $\square$
