Screenshot of a result with sketch of its proof
I have understood the parts above the red arrow. I'm stuck after that. While I can intuitively understand that the difference between the order statistic and $F^{-1}[Y_n(x)]$ converges almost surely to $0$, I think that it is not enough to get the desired result and still needs a mathematical proof. However, my intuition is that the difference between the order statistic and $F^{-1}[Y_n(x)]$ is the order of $(1/n)$. Can anyone give a mathematical proof for the same?