# Proving uniform convergence of moment restriction score function in GMM asymptotic normality proof

I am asked in a homework question to prove asymptotic normality for the generalized method of moments estimator. The assumptions (which i think are necessary to solve this particular subproblem) given in the theorem are

1. $$(Z_i)_{i \in \mathbb{N}}$$ is a sequence of i.i.d. random variables.
2. $$g(z|\theta)$$ is continuously differentiable wrt. $$\theta$$ in a neighborhood $$\mathcal{N}$$ of $$\theta_0\in Int(\Theta)$$ ($$g$$ is a moment restriction function, $$\theta_0$$ is the true parameter, and $$\Theta\subset\mathbb{R}^k$$ is the parameter space)
3. $$\mathbb{E}[\sup_{\theta\in \mathcal{N}}||g(Z_i|\theta)||]<\infty$$

In the concluding argument of the proof I need to show that $$G_n(\theta):=\frac{1}{n}\sum_{i=1}^{n}\partial_\theta g(Z_i|\theta)$$ converges uniformly to $$G(\theta) = \mathbb{E}[\partial _\theta g(Z_i|\theta)]$$, i.e. $$\sup_{\theta\in\mathcal{N}} ||G_n(\theta) - G(\theta)||\stackrel{P}{\rightarrow}0$$ It is hinted at that the convergence follows from conditions 2 and 3. I have also snooped around various stack exchanges and gotten a hunch that the Borel-Cantelli lemma might be helpful. But at this point I am truly lost.

Any help would be greatly appreciated!