I see that, but the results on page 9 and 14 are given for the finite counting process, though on arbitrary intervals. Are you saying that a suitable choice of the interval plus proper scaling can recover the GOE/GUE point process? I know that $\{ \frac{\lambda_{N-k} - 2\sqrt{N}}{N^{1/6}} \}_{k=0}^{k_0} \xrightarrow{d} \{a_k \}_{k=0}^{k_0} $, but this convergence is finite dimensional.

I believe the results on page 9 and 14 hold for the bulk eigenvalue point process. The GOE and GUE point process refers to the edge-scaled eigenvalue point process.