$\def\bbR{\mathbb{R}}$I have looked for a while about this in MSE and I couldn't find anything. Here it goes: In J. M. Lee, Introduction to Riemannian Manifolds, 2nd ed., there is this result:
I wanted to ask about Lee's proof of Theorem 7.11. For each $\delta,\varepsilon\in\bbR\setminus\{0\}$ and $z\in T_pM$, denote the limiting term in the RHS of (7.10) as $$ F_{\delta,\varepsilon}(z)=\frac{P_{\delta, 0}^{0,0} \circ P_{\delta, \varepsilon}^{\delta, 0} \circ P_{0, \varepsilon}^{\delta, \varepsilon} \circ P_{0,0}^{0, \varepsilon}(z)-z}{\delta \varepsilon}. $$ What Lee's proof actually does is showing that the following iterated limit formula holds: $$ R(x,y)z= \lim_{\delta\to 0}\lim_{\varepsilon\to 0}F_{\delta,\varepsilon}(z). $$ My question is: how does one show then that $\displaystyle\lim_{\delta\to 0}\lim_{\varepsilon\to 0}F_{\delta,\varepsilon}(z)=\lim_{\delta,\varepsilon\to 0}F_{\delta,\varepsilon}(z)$?
Looking "iterated limits" in google brought me to this:
I am trying to check the hypotheses for our situation. On the one hand, from Lee's proof one has that $\displaystyle\lim_{\varepsilon\to 0}F_{\delta,\varepsilon}(z)$ exists. But on the other hand, I don't know how to show existence of $\displaystyle\lim_{\delta\to 0}F_{\delta,\varepsilon}(z)$, nor how to argue that of among the limits $\displaystyle\lim_{\varepsilon\to 0}F_{\delta,\varepsilon}(z)$ and $\displaystyle\lim_{\delta\to 0}F_{\delta,\varepsilon}(z)$, at least one converges uniformly.
My questions are:
How one would verify the other hypothesis of the theorem on interchange of two limits?
Is there any different approach to this auxiliar theorem to show that $\displaystyle\lim_{\delta\to 0}\lim_{\varepsilon\to 0}F_{\delta,\varepsilon}(z)=\lim_{\delta,\varepsilon\to 0}F_{\delta,\varepsilon}(z)$?
In case it helps, here is the expression from Lee's proof for the limit in $\varepsilon$: $$ \lim_{\varepsilon\to 0}F_{\delta,\varepsilon}(z) = \frac{P_{\delta, 0}^{0,0}\left(D_t Z(\delta, 0)\right)-D_t Z(0,0)}{\delta}, $$ where $D_t$ is the covariant derivative along the curve $t\mapsto\Gamma(s,t)$ (where $s=\delta$ on the first case and $s=0$ in the second case) and $Z$ is the vector field along $\Gamma$ (i.e., a lifting to $TM\to M$ of the map $\Gamma$) explained at the beggining of Lee's proof:
Define a vector field $Z$ along $\Gamma$ by first parallel transporting $z$ along the curve $t\mapsto\Gamma(0,t)$, and then for each $t$, parallel transporting $Z(0,t)$ along the curve $s\mapsto \Gamma(s,t)$. The resulting vector field along $\Gamma$ is smooth by another application of Theorem A.42 (the fundamental theorem on flows) as in the proof of Lemma 7.8.
In the quote I have incorporated the correction to p. 201.