Finite bracketing integral implies convergence in probability

In a paper I am reading, the following result is used. If $$(\mathcal{X},\mathcal{F})$$ is a measurable space, $$\mathcal{G}$$ is a class of functions with elements $$f : \mathcal{X} \to \mathbb{R}$$. We assume $$\mathcal{G}$$ is a bounded metric space. Then I wish to show that if

$$\int_{0}^R\sqrt{H_B(u,\mathcal{G},L_2(P))}du < \infty$$

where $$H_B(u,\mathcal{G},L_2(P))$$ is the bracketing entropy, then

$$\sup_{f \in \mathcal{G}} \Big| \frac{1}{n}\sum_{i=1}^nf(X_i)-\mathbb{E}(f(X)) \Big| \to 0$$ in probability. Here, $$R$$ is a bound on the distance between functions in $$\mathcal{G}$$, i.e.

$$\sup_{f_1,f_2 \in \mathcal{G}} d(f_1,f_2) < R.$$

I have looked various resources on empirical process theory, for example page 50 of http://www.stat.columbia.edu/~bodhi/Talks/Emp-Proc-Lecture-Notes.pdf#page50, but to no avail. The paper where it has been used is page 3 of https://projecteuclid.org/journals/supplementalcontent/10.1214/14-AOS1260/suppdf_1.pdf