I'm reading the John Lee's Introduction to Riemannian manifold, p.333, p.335 and some question arises
First of all, let's see Lemma 11.13 in his book p.333
Lemma 11.13. Suppose $(M,g)$ is a Riemannian manifold and $(x^i)$ are Riemannian normal coordinates on a normal neighborhood $U$ of $p\in M$. Let $\det g$ denote the determinant of the matrix $(g_{ij})$ in these coordinates, let $r$ be the radial distance function, and let $\partial_r$ be the unit radial vector field. The following identity holds on $U-\{p\}$ : $$ \Delta r = \partial_r \log ( r^{n-1} \sqrt{\det g}). \tag{11.23}$$
Q. (How) can we view the $\sqrt{\det g}$ as a function of $r$?
Second, In his book, p.335, ( proof of the Gunther's Volume Comparison ), he wrote :
"Note that $g_{ij}= \delta_{ij} $ at the origin ( In the second paragraph in the proof of the Gunther's Volume Comparison, $(x^i)$ is a normal coordinates on some geodesic ball $B_{\delta_0}(p)$ and he saids that "we might as well consider $g$ to be a Riemannian metric on an open subset of $\mathbb{R}^n$ and $p$ to be origin) , so $\sqrt{\det g}$ converges uniformly to $1$ as $r\searrow 0$"
Why the statement(uniform convergence) is true? If more information is needed, then I will upload.
EDIT : As a result of some consideration, perhaps, that "(restriction to $U$ of) $\sqrt{\det g}$ is a function $U \to \mathbb{R}$" is true and the statement "$\sqrt{ \det g}$ converges to $1$ and $r \searrow 0$ uniformly" means that $\sqrt{\det g}$ converges to $1$ as points of $U$ approaches to the origin in any direction ? In particular, if $(x_n)$ is a sequence of $U$ which converges to the origin, then $\sqrt{\det g(x_n)}$ converges to $1$?
Reason of interpreting like this orginates from following Theorem (John Lee's book Corollary 6.12 ):
Corollary 6.12. Let $(M,g)$ be a connected Riemannian manifold and $p\in M$. Within every open or closed geodesic ball around $p$, the radial distance function $r(x)$ defined by $(6.4)$ is equal to the Riemannian distance from $p$ to $x$ in $M$.
Am I understanding well? If so, intuitively it(the uniform convergence of the above) seems to be true and I can't prove that more rigorously.
Can anyone helps?