Reference request: Lie product formula for vector fields

Question

Let $K$ be a compact neighborhood in $\mathbb R^n$, $Z=X+Y$ are (non-vanishing if necessary) smooth vector fields on $K$. Denote by $e^{sZ}p$ an integral curve of $Z$ with initial point $p=e^{0Z}p\in K$ and a terminal point $q=e^{tZ}p\in K$. Then for any $\varepsilon>0$ there exists a large enough natural number $N$ such that

$$||(e^{\frac{s}{N}X}e^{\frac{s}{N}Y})^Np - e^{sZ}p||<\varepsilon$$

for all $0<s<t$.

I am looking for a reference for this statement (preferably a textbook)

Answer 1

Abraham and Marsden, Foundations of Mechanics, p. 78, Corollary 2.1.27; Let $X, Y$ be vector fields on a manifold $M$ with flows $F_t$, $G_t$. Let $H_t$ be the flow of $X+Y$. Then $$ H_t(x) =\lim_{n \to \infty} (F_{t/n} \circ G_{t/n})^n $$ Each side is defined if and only the other is.