Vaart, A. W. van der (1998), Asymptotic statistics, Cambridge series in statistical and probabili...

Vaart, A. W. van der (1998), Asymptotic statistics, Cambridge series in statistical and probabilistic mathematics, Cambridge, UK ; New York, NY, USA: Cambridge University Press. Exercise 18.6

It suffice to prove for . For any , let

Let . Because is right-continuous at , we have . Then and is an upper bound of . Let . By the supremum and infimum principle, it suffices to prove and .

Obviously, we have . Because has left limit in , there exist such that for any , . For , by definition of supremum, there exist such that . Without loss of generality, assume . Let . Hence for any , and , we have . It follows .

As for , noting that if , we can find such that and . This contracts the fact .

Combining and , we complete the proof.