Let $(M,g)$ be an arbitrary $2$-dimensional Riemannian manifold. For the meaning of a $\varepsilon$-net on $M$ I refer to the wikipedia-page. When $X_{\varepsilon}\subseteq M$ is an epsilon net and $x\in X_{\varepsilon}$, let me recursively define $\text{Cl}^j(x)\subseteq X_{\varepsilon}$ as $\text{Cl}^0(x):=\{x\}$ and $$\text{Cl}^{j+1}(x):=X_{\varepsilon}\cap\left(\bigcup_{z\in \text{Cl}^{j}(x)}B(z,2\varepsilon)\right).$$ Is it so that $$\lim_{\varepsilon\downarrow 0}\,\,\inf_{X_{\varepsilon}}\,\,\inf_{x\in X_{\varepsilon}}\sum_{j=0}^{\lceil \text{diam}(M)/(2\varepsilon)\rceil} \frac{1}{|\text{Cl}^{j+1}(x)\setminus\text{Cl}^j(x)|}=+\infty?$$ (For the occasion, we use the unusual convention $\frac{1}{0}=0$)
(The reason why I suspect so is that the summands in the sum are expected behave like $|\text{Cl}^{j+1}(x)\setminus\text{Cl}^j(x)|=O(j)$ and the harmonic series diverges?)
