Definitions: Consider a random variable $X:\Omega \rightarrow \mathcal{X}$ defined on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ with probability distribution $P$. All functions mentioned from now on will be random functions from $\mathcal{X}$ to $\mathbb{R}$. Consider two functions $l,u$. A bracket $[l,u]$ is the set of all functions $f$ with $l\leq f\leq u$. A $\epsilon$-bracket in $L_r(P)$ is a bracket with $\int_{\mathcal{X}}(u-x)^rdP<\epsilon^r$ with $0<\epsilon<\infty$ and $r>0$. The $L_r(P)$ norm of a function $f$ is $(\int_{\mathcal{X}}|f|^rdP)^{\frac{1}{r}}$. Let $\mathcal{F}$ be a class of functions $f$. The bracketing number $N_{[\text{ }\text{ }]}(\epsilon, \mathcal{F}, L_r(P))$ is the minimum number of $\epsilon$-brackets in $L_r(P)$ to cover $\mathcal{F}$. The entropy is $\log(N_{[\text{ }\text{ }]}(\epsilon, \mathcal{F}, L_r(P)))$. Notice that $N_{[\text{ }\text{ }]}(\epsilon, \mathcal{F}, L_r(P))$ and $\log(N_{[\text{ }\text{ }]}(\epsilon, \mathcal{F}, L_r(P)))$ are decreasing in $\epsilon$.
Questions: If $\log(N_{[\text{ }\text{ }]}(\epsilon, \mathcal{F}, L_r(P)))\in O(\frac{1}{\epsilon^2})$ then $N_{[\text{ }\text{ }]}(\epsilon, \mathcal{F}, L_r(P))<\infty$? I.e. if $\mathcal{F}$ is a Glivenko-Cantelli class then it is also a Donsker class? Why? (from p.270 van der Vaart "Asymptotic Statistics")