Uniform convergence of some functions ( John Lee's Introduction to Rieamnnian manifold, Gunther's Volume Comparison )

Question

I am reading the John Lee's Introduction to Rieamnnian manifold, proof of Theorem 11.14 (Gunther's Volume Comparison ) and stuck at some point. I posted question including next questions :Understanding the Gunther's Volume comparison theorem ( John Lee's Introductino to Riemannian manifold ) (C.f. questions Q.3-1) and Q,4 in the linked question), and its length seems to be somewhat long and so I separately ask questions like this. Anyway so, in this question, I upload associated material (its proof) as a image.

I don't understand the underlined statement.

There, in the proof, $s_c : \mathbb{R} \to \mathbb{R}$ defined by $(10.8)$ (in his book) is, for each $c\in \mathbb{R}$,

$$ s_c(t) = \begin{cases} t, & \mbox{if }\mbox{ c=0} \\ R\sin \frac{t}{R}, & \mbox{if }\mbox{$c=\frac{1}{R^2} >0$} \\ R\sinh \frac{t}{R}, & \mbox{if }\mbox{ $c= - \frac{1}{R^2} <0$ } \end{cases} \tag{10.8} $$

In Corollary 6.12 in John Lee's book, he states

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 $M$.

So I understood the statement ( marked by red in the above image ) $r^{n-1} \sqrt{\det g}/s_c(r)^{n-1} \to 1$ as follows. Note that $r^{n-1} \sqrt{\det g}/s_c(r)^{n-1}$ is a function from $U:= B_{\delta_0}(p \stackrel{\text{?}}{=}0) $ to $\mathbb{R}$. Then the above convergence statement implies that for each sequence $(x_n) \to 0$ ( in $U = B_{\delta_0}(p) )$,

$$ r(x_n)^{n-1} \sqrt{\det g(x_n)}/s_c(r(x_n))^{n-1} \xrightarrow{ n \to \infty} 1 \tag{1}$$

(Notational Caution : The superscript $n-1$ is about the dimension $n$ of the manifold, not related to the subscripts of $x_n$.) True?

Can we extract more information from the above convergence $r^{n-1} \sqrt{\det g}/s_c(r)^{n-1} \to 1$?

(C.f. This question is related to other question About determinant of metric tensor and the radial distance function on a normal neighborhood. (And furthur question ) that I also proposed. )

As the underlined statement, to show the uniform convergence of $\lambda(\rho, \omega)$ to $1$ as $\rho \searrow 0$, choose $\rho_n \searrow 0$ with $\rho_n \in (0, \delta_0)$. We try to show that the functions $\lambda_n(w) := \lambda(\rho_n , \omega) : \mathbb{S}^{n-1} \to \mathbb{R}$ converges uniformly to $1$.

Note that $r(\rho \omega)$, by the above Corolary 6.21 which is the Riemannian distance from the $p=0$ to $\rho \omega$, is $\rho$. I think that we can show this by noting that $\omega \in \mathbb{S}^{n-1}$ (distance 1). True?

Note that

$$ | \lambda_n(\omega) -1 | := |\lambda( \rho_n, \omega) -1| = | \rho_n^{n-1} \sqrt{\det g(\rho_n \omega)}/s_c(\rho_n)^{n-1} -1| \\ = |r(\rho_n \omega)^{n-1} \sqrt{\det g(\rho_n \omega)}/s_c(r(\rho_n \omega))^{n-1} -1 | \xrightarrow{ n \to \infty} 0, \tag{2}$$

by the above $(1)$.

So $\lambda_n(\omega)$ converges to $1$ pointwise. And is it also converges uniformly? Such convergence does not depend on the $\omega$? How to show? What should I catch?

My first strategy is, showing follows :

• 1. If $f_{n}(\omega) \to f(\omega)$ and $g_n(\omega) \to g(\omega)$ are real-valued uniformly convergent functions on $\mathbb{S}^{n-1}$ then $f_n \cdot g_n$ converges uniformly to $f\cdot g$.
• 2. $\sqrt{\det g ( \rho_n \omega)}$ converges to $1$ uniformly, as the part marked by red colour in the above image.
• 3. $\frac{r(\rho_n \omega)^{n-1}}{s_c(r(\rho_n \omega))^{n-1}}$ converges to $1$ uniformly. Note that since $r( \rho_n \omega) = \rho_n$, it suffices to show that $\frac{\rho_n^{n-1}}{s_c(\rho_n)^{n-1}}$ converges to $1$ as $n \to \infty$. The sufficiency is true?

and we are done. Correct argument? These are true?

Again, can we extract more information from $r^{n-1} \sqrt{\det g}/s_c(r)^{n-1} \to 1$ ?

P.s. I think that perhaps, the argument about showing $r^{n-1} \sqrt{\det g}/s_c(r)^{n-1} \to 1$ ( marked by red ) is redundant. Reason of this is, since usage of the argument marked by red is at last to show the uniform convergence of $\lambda(\rho, \omega)$ as $\rho \searrow 0$ ( if we look the whole proof somewhat closerly ), and its proof can be proven directly if our above three questions are true, without going through the "$r^{n-1} \sqrt{\det g}/s_c(r)^{n-1} \to 1$".

Can anyone helps?