Im trying to prove a variation of Harnack's theorem want to show that a sequence of positive harmonic functions $h_n:G\to \mathbb{R}$ diverge locally uniformly to infinity if they diverge to infinity pointwise.
For a fixed $w$ I have deduced this inequality: $$\tau_G(z,w)^{-1}h_n(w)\leq h(z)$$ where the the Harnack distance $\tau_G(z,w)$ is defined as the smallest number such that $\tau_G(z,w)^{-1}h(w)\leq h(z)\leq \tau_G(z,w)h(w)$ for all positive harmonic functions $h$ on $G$. so if we assume that $h_n(w)\to \infty$ does that mean $h(z)\to \infty$ locally uniformly?