Let $\mathbf{G}$ be a square matrix of dimension $N$ with zeros on the diagonal. I assume that the entries of $\mathbf{G}$ are in the interval $[0, 1$] and $\displaystyle \lim_{N\to\infty}\sum_{j = 1}^N\mathbf{G}_{ij} = 1$.
I want to prove that $$\lim_{N \to \infty} \mathbf{Trace}\left((\mathbf{I} - \alpha \mathbf{G})^{-1}\mathbf{G}\right) < \infty,$$ for any $\alpha$ in $(0, 1)$, where $\mathbf{I}$ is the identity matrix of dimension $N$.
This is also equivalent to prove that
$$\lim_{N \to \infty} \mathbf{Trace}\left(\sum_{k = 0}^{\infty} \alpha^k\mathbf{G}^{k + 1}\right) < \infty.$$