Approximation by finite subsets and strong resolvent convergence

Question

Let $\mathbb{G}$ be an at most countable set (e.g., $\mathbb{G}=\mathbb{Z}^d$) and $H$ be a self-adjoint operator (not necessarily bounded) on $l^2(\mathbb{G})$. Let $\mathbb{G}_L$ be finite subsets of $\mathbb{G}$ such that $\mathbb{G}_L\nearrow \mathbb{G}$ and $H_L=P_L H P_L$ where $P_L$ are orthogonal projections onto $\mathbb{G}_L$ (i.e., if $|x\rangle $ denote the dirac functions localized at $x\in \mathbb{G}$, then we can write $P_L = \sum_{x\in \mathbb{G}_L} |x\rangle \langle x|$). Then Aizneman (in Random Operators, Chapter 7.2) claims that $H_L \to H$ in strong resolvent sense, i.e., $(H_L -z)^{-1} \to (H-z)^{-1}$ in strong operator topology for all $\Im{z} \ne0 $. However, he doesn't provide a proof, so I can't quite understand why this is true.