I have some doubts related to example 19.7 in van der Vaart "Asymptotic Statistics" which applies Theorem 19.4 (Glivenko-Cantelli) and Theorem 19.5 (Donsker) to a parametric class of functions.
Setting:
(i) $\{X_i\}_{i=1}^N$ i.i.d. defined on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$; $X_i:\Omega \rightarrow \mathcal{X}$; $P$ is the probability distribution function of $X_i$
(ii) Take the the class of functions $\mathcal{F}:=\{f_\theta \text{ s.t. } f_\theta: \mathcal{X} \rightarrow \mathbb{R} \text{ and } \theta \in \mathbb{\Theta} \subseteq \mathbb{R}^l\}$
(iii) $\exists$ $m:\mathcal{X}\rightarrow \mathbb{R}$ such that $|f_{\theta_1}(x)-f_{\theta_2}(x)|\leq m(x)||\theta_1-\theta_2||$ $\forall \theta_1, \theta_2\in \Theta$ and $\forall x \in \mathcal{X}$
(iv) $\int_{\mathcal{X}}|m(x)|^rdP< \infty$ for some $r>0$
(v) $||m||_{P,r}:=(\int_{\mathcal{X}}|m(x)|^rdP)^{\frac{1}{r}}$
I want to show that
(1) $N_{[ \text{ }]}(\epsilon, \mathcal{F}, L_1(P))<\infty$ $\forall \epsilon>0$
(2) $J_{[ \text{ }]} (1, \mathcal{F}, L_2(P))<\infty$ $\leftrightarrow$ $\log N_{[ \text{ }]}(\epsilon, \mathcal{F}, L_1(P))\in O(\frac{1}{\epsilon^2})$
Here a summary of the proof in the book with my questions:
(a) I believe that (2) implies (1). Hence the proof is focused on showing (2)
(b) Do we need $r\geq 2$ to proceed? I believe that $\int_{\mathcal{X}}|m(x)|^rdP< \infty$ for $r\geq 2$ implies $\int_{\mathcal{X}}(m(x))^2dP< \infty$. In what follows I proceed with $r=2$ even if in the book the author uses $r$
(c) We can show that $\exists$ $0<K<\infty$ such that $1\leq N_{[ \text{ }]}(\epsilon||m||_{P,2}, \mathcal{F}, L_2(P))\leq K*(\frac{diam(\Theta)}{\epsilon})^l$ $\forall$ $0<\epsilon<diam(\Theta)$
(d) Hence $\log (N_{[ \text{ }]}(\epsilon, \mathcal{F}, L_2(P)))$ is of order smaller then $\log(\frac{1}{\epsilon})$ which implies $\log (N_{[ \text{ }]}(\epsilon, \mathcal{F}, L_2(P))) \in O(\frac{1}{\epsilon^2})$.
In fact, (c) implies $\forall$ $0<\epsilon<diam(\Theta)$ $$0\leq \log(N_{[ \text{ }]}(\epsilon||m||_{P,2}, \mathcal{F}, L_2(P))\leq \log(K)+l\log(\frac{diam(\Theta)}{\epsilon})$$
Moreover, $\log(K)+l\log(\frac{diam(\Theta)}{\epsilon})\in O(\log(\frac{1}{\epsilon})) $ as $\epsilon \rightarrow 0$ which means means that $\exists 0<M<\infty$, $\eta>0$ such that $\forall 0<\epsilon<\eta$ $$ \log(K)+l\log(\frac{diam(\Theta)}{\epsilon})\leq M \log(\frac{1}{\epsilon})$$
Hence, $\forall 0<\epsilon<\min\{\eta, diam(\Theta)\}$
$$
0\leq \log(N_{[ \text{ }]}(\epsilon||m||_{P,2}, \mathcal{F}, L_2(P))\leq M \log(\frac{1}{\epsilon})
$$
Moreover if $||m||_{P,2}<1$ (is this the case? I think it should be also looking at p.77 last line of the proof of Corollary 5.53) then $\forall 0<\epsilon<\min\{\eta, diam(\Theta)\}$
$$
0\leq \log(N_{[ \text{ }]}(\epsilon, \mathcal{F}, L_2(P)) \leq \log(N_{[ \text{ }]}(\epsilon||m||_{P,2}, \mathcal{F}, L_2(P))\leq M \log(\frac{1}{\epsilon})
$$
Hence,
$$
\log(N_{[ \text{ }]}(\epsilon, \mathcal{F}, L_2(P))\in O(\log(\frac{1}{\epsilon}))\in O(\frac{1}{\epsilon^2})
$$
Any hint on the topic would be really appreciated.