This is actually a detail in Lee’s smooth manifold 2nd ed p.234, 2nd paragraph in the Theorem 9.46. What i found is possibly a correction for the book. Forgive me if it’s not even close.
Let $M$ be a smooth manifold of dimension $n$ and $p \in M$ be an arbitrary point and $U$ is a domain of a smooth chart centered at $p$. Suppose that we have $k$-tuple of smooth vector fields $(V_1, \dots, V_k)$ defined on $U$. Let $\theta_i$ denote the flow of $V_i$.
Before i state the problem that has bugged me, i want to note that actually; first, the chart $U$ above is a slice chart. Second the vector fields $(V_i)$ above is linearly independent and mutually commuting. But for the following problem, i think we will not use those assumptions. Here’s the problem
Lee claim that there exists $\epsilon>0$ and a neighbourhood $Y$ of $p$ in $U$ such that the composition $(\theta_1)_{t_1} \circ (\theta_2)_{t_2} \circ \cdots \circ (\theta_k)_{t_k}$ defined on $Y$ and maps $Y$ into $U$ whenever $|t_1|,\dots,|t_k|$ are all less than $\epsilon$.
After this claim, he also said that
To see this, just choose $\epsilon_k>0$ and $U_k \subseteq U$ such that $\theta_k$ maps $(-\epsilon_k,\epsilon_k)\times U_k$ into $U$, and then inductively choose $\epsilon_i$ and $U_i$ such that $\theta_i$ maps $(-\epsilon_i,\epsilon_i)\times U_i$ into $U_{i+1}$. And then taking $\epsilon = \text{min }\{\epsilon_i\}$ and $Y=U_1$.
I did the suggested construction but i think the suggestion should be
To see this, just choose $\epsilon_1>0$ and $U_1 \subseteq U$ such that $\theta_1$ maps $(-\epsilon_1,\epsilon_1)\times U_1$ into $U$, and then inductively choose $\epsilon_{i}$ and $U_{i}$ such that $\theta_{i}$ maps $(-\epsilon_{i},\epsilon_{i})\times U_{i}$ into $U_{i-1}$. And then taking $\epsilon = \text{min }\{\epsilon_i\}$ and $Y=U_k$.
I may get this wrong. I hope somebody could clarify this for me. Is the suggestion is correct or it should be other way around as i did. Any help will be appreciated. Thank you