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
![\begin{align*} A& :=\left\{z \in[-u, u]: \exists \delta>0, m\in\mathbb{N}, u_i\in[-u,z], i = 1,\ldots, m, s.t.,\right.\\ & \left. \bigcup_{i = 1}^{m - 1}[u_i, u_{i + 1}) = [-u,z), \forall x, y \in [u_i,u_{i+1}), |x-y|<\delta, |h(x) - h(y)|<\epsilon\right\} \end{align*}](https://math.jianshu.com/math?formula=%5Cbegin%7Balign*%7D%20A%26%20%3A%3D%5Cleft%5C%7Bz%20%5Cin%5B-u%2C%20u%5D%3A%20%5Cexists%20%5Cdelta%3E0%2C%20m%5Cin%5Cmathbb%7BN%7D%2C%20u_i%5Cin%5B-u%2Cz%5D%2C%20i%20%3D%201%2C%5Cldots%2C%20m%2C%20s.t.%2C%5Cright.%5C%5C%20%26%20%5Cleft.%20%5Cbigcup_%7Bi%20%3D%201%7D%5E%7Bm%20-%201%7D%5Bu_i%2C%20u_%7Bi%20%2B%201%7D)%20%3D%20%5B-u%2Cz)%2C%20%5Cforall%20x%2C%20y%20%5Cin%20%5Bu_i%2Cu_%7Bi%2B1%7D)%2C%20%7Cx-y%7C%3C%5Cdelta%2C%20%7Ch(x)%20-%20h(y)%7C%3C%5Cepsilon%5Cright%5C%7D%20%5Cend%7Balign*%7D)
Let %3A%3D%5C%7B%5Bu_i%2Cu_%7Bi%2B1%7D)%3A%20i%20%3D%201%2C%5Cldots%2C%20m%5C%7D)
. 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 ![c\in[-u, u]](https://math.jianshu.com/math?formula=c%5Cin%5B-u%2C%20u%5D)
. Because has left limit in 
, there exist 
such that for any )
, %20-%20h(y)%7C%3C%5Cepsilon)
. For 
, by definition of supremum, there exist 
such that 
. Without loss of generality, assume 
. Let %3A%3D%5Cmathcal%7BU%7D(x)%5Ccup%20%5B%5Calpha%2C%20c))
. Hence for any )
, and 
, we have %20-h(y)%7C%3C%5Cepsilon)
. It follows 
.
As for , noting that if 
, we can find 3%2F4)
such that %2F2)
and 
. This contracts the fact 
.
Combining and , we complete the proof.
