$G_{n0}\vert _{n\rightarrow \infty } \rightarrow 0$?

This is not true. Consider the case where there exists a single site in the unit cel and $H_{00}=0$ (energy level) and $H_{10}=t$ (transfer integral to the nearest neighbor site).

$$G_{2n,0}\vert _{n\rightarrow\infty}(\omega ) \sim t^{2n-1}/\mbox{Det}\left(\omega I -H\right)$$ (54)

If $\omega =0$, or $\omega$ is equal to the energy level,

$$\mbox{Det}\left(\omega I -H\right) = (-1)t^{2n}$$ (55)

(Its matrix size is $2n$.) Then,

$$G_{2n,0}(\omega =0) \vert _{n\rightarrow\infty} \sim 1/t$$ (56)