Exercise (HW1): Let $\mathcal{F}$ and $\mathcal{G}$ be classes of measurable function. Then for any probability measure $Q$ and any $1 \leq r \leq \infty$, (i) $N_{[]}\left(2 \epsilon, \mathcal{F}+\mathcal{G}, L_{r}(Q)\right) \leq N_{[]}\left(\epsilon, \mathcal{F}, L_{r}(Q)\right) N_{[]}\left(\epsilon, \mathcal{G}, L_{r}(Q)\right)$; (ii) provided $\mathcal{F}$ and $\mathcal{G}$ are bounded by 1 , $$ N_{[]}\left(2 \epsilon, \mathcal{F} \cdot \mathcal{G}, L_{r}(Q)\right) \leq N_{[]}\left(\epsilon, \mathcal{F}, L_{r}(Q)\right) N_{[]}\left(\epsilon, \mathcal{G}, L_{r}(Q)\right) $$

This is the question. I need to proof part (ii) for my work.**Assume those Ns are coverning numbers and not bracketing numbers as he considers**

Reference. Question is taken from page 21,2nd HW1 part (ii)

I am getting no clue. Can anyone help me in this?

$2.2$ Bracketing numbers Let $(\mathcal{F},\|\cdot\|)$ be a subset of a normed space of real functions $f: \mathcal{X} \rightarrow \mathbb{R}$ on some set $\mathcal{X}$. We are mostly thinking of $L_{r}(Q)$-spaces for probability measures $Q$. We shall write $N\left(\varepsilon, \mathcal{F}, L_{r}(Q)\right)$ for covering numbers relative to the $L_{r}(Q)$-norm $\|f\|_{Q, r}=\left(\int|f|^{r} d Q\right)^{1 / r}$. Definition $2.9$ ( $\varepsilon$-bracket). Given two functions $l(\cdot)$ and $u(\cdot)$, the bracket $[l, u]$ is the set of all functions $f \in \mathcal{F}$ with $l(x) \leq f(x) \leq u(x)$, for all $x \in \mathcal{X}$. An $\varepsilon$-bracket is a bracket $[l, u]$ with $\|l-u\|<\varepsilon$.

Definition 2.10 (Bracketing numbers). The bracketing number $N_{[]}(\varepsilon, \mathcal{F},\|\cdot\|)$ is the minimum number of $\varepsilon$-brackets needed to cover $\mathcal{F}$.

Definition $2.11$ (Entropy with bracketing). The entropy with bracketing is the logarithm of the bracketing number.

In the definition of the bracketing number, the upper and lower bounds $u$ and $l$ of the brackets need not belong to $\mathcal{F}$ themselves but are assumed to have finite norms.