Suppose $M$ is a compact $n$-dimensional Riemannian manifold. I know that for every $p \in M$ it holds $$ \tag{1} \frac{\operatorname{Vol}(B_\varepsilon(p)\subset M)}{\operatorname{Vol}(B_\varepsilon(0)\subset {\mathbb R}^n)}=1-\frac{S(p)}{6(n+2)}\varepsilon^2 + O(\varepsilon^4)$$ where $S$ is the scalar curvature of $M$.
Since the manifold is compact and $S$ is continuous, there exists some $S_0 >0$ s.t $|S(p)| \le S_0$ for all $p \in M$.
Does this allows me to conclude that:
There exist $\delta= \delta(M)>0$ and $C_1=C_1(M)>0, C_2=C_2(M)>0$ s.t. $$ C_1 r^n \le \operatorname{Vol}(B_\varepsilon(p)\subset M) \le C_2 r^n $$ for every $p \in M$ and $0<r \le \delta$.
?
I am not very sure about that since $(1)$, without the big-O notation, says that
For every $p \in M$ there exist $\delta_p >0 $ and a constant $C_p>0$ s.t.
$$ \Biggl | \frac{\operatorname{Vol}(B_\varepsilon(p)\subset M)}{\operatorname{Vol}(B_\varepsilon(0)\subset {\mathbb R}^n)}-1+\frac{S(p)}{6(n+2)}\varepsilon^2 \Biggr | \le C_p \epsilon^4 $$
for every $\epsilon < \delta_p$.
The problem is that I don't know if $\delta_p$ and $C_p$ are continuous functions of $p$ and then it may be that $\inf_p \delta_p =0$ and $\sup_p C_p = + \infty$.
Is there any way to solve this issue?