Let $\{Z_i\}_{I=1}^n$ be i.i.d. random variables, $\{\delta_n\}$ be a sequence of positive numbers such that $\delta_n\rightarrow 0$, and $||g||_{\mathcal{G}}=\sup_z|g(z)|$, then
\begin{equation*}
\sup_{||g||_{\mathcal{G}}\leq \delta_n}\left|\frac{1}{\sqrt{n}}\sum_{i=1}^n (g(Z_i)-E[g(Z_i)])\right|\rightarrow_{a.s.} 0.
\end{equation*}
I need to prove the result above, and have freedom to make some assumptions. I can suppose that any moments of $Z_i$ and $g(Z_i)$ are bounded. I can also make assumptions on $\mathcal{G}$ (uniformly bounded, continuous, smooth, Donsker class, bounds on covering numbers etc.) I cannot suppose that $\delta_n$ converges to zero at a specific rate.
This result can be shown for the convergence in probability with second moment bounds, but I am having difficulty showing it for almost sure convergence. Can you help me?