The proof can be found in [1]. The main idea is to construct a cover by choosing point \(\{p_i\}\) such that the distance between two distinct points is strictly larger then \(\frac{\varepsilon}{2}\) and \(\dist(\partial B_r(p), p_i)>\frac{\varepsilon}{2}\).

Using triangle inequalities we see that \(\{B_{\varepsilon}(p_i)\}\) covers \(B_r(p)\). Furthermore, \(\{B_{\varepsilon/4}(p_i)\}\) are disjoint, such that

\begin{equation*} \lvert \{p_i\}\rvert\le \sup_{i} \frac{\Vol(B_r(p))}{\Vol(B_{\varepsilon/4}(p_i))} \le \sup_{i} \frac{\Vol(B_{2r}(p_i)}{\Vol(B_{\varepsilon/4}(p_i))}. \end{equation*}

Applying Bishop-Gromov inequality we obtain the first bound.

To receive the second bound, we consider points \(p_j\) for a given point \(p_i\) such that \(B_{\varepsilon}(p_j)\cap B_{\varepsilon}(p_i)\neq \emptyset\).

Then, \(B_{\varepsilon/4}(p_j)\subset B_{3\varepsilon}(p_i)\) and similarly to the argument before we get by applying again Bishop-Gromov inequality

\begin{equation*} \lvert \{p_j\mid B_{\varepsilon}(p_j)\cap B_{\varepsilon}(p_i)\neq \emptyset,\; i\neq j \}\rvert \le \sup_i \frac{\Vol(B_{3\varepsilon}(p_i))}{\Vol(B_{\varepsilon/4}(p_i))}\le C_2. \end{equation*}