# On Spivak's introduction to one parameter group of diffeomorphisms

In Chapter 5 of M. Spivak's A Comprehensive Introduction To Differential Geometry he states the existence and uniqueness of integral curves $$\alpha_x:(-b,b)\rightarrow U$$ of an ODE with initial condition $$\alpha_x'(t)=f(\alpha(t))$$ $$\alpha_x(0)=x$$ This is proven for a map $$f:U\rightarrow \mathbb{R}^n$$ defined in an open set $$U$$ and every $$x\in B_a(x_0)$$ for some $$a>0$$ and $$x_0\in U$$ such that the closed ball $$\overline{B}_{2a}(x_0)\subset U$$, we also ask $$f$$ to be locally Lipschitz (there are as well some restriction on $$b$$). Next he introduces the flow $$\alpha:(-b,b)\times B_a(x_0)\rightarrow U\quad \;\text{as }\quad (t,x)\mapsto\alpha_x(t)$$ Ok! this is all great and fairly straightforward, but the next discussion has an important step that I do not follow, here it is:

He argues that $$\alpha$$ is continuous (proven, no problem!) and since every $$\alpha_y$$ satisfies the condition $$\alpha_y(0)=y$$, then we have $$\alpha:\{0\}\times \overline{B} _{a/2}(x_0)\rightarrow \overline{B} _{a/2}(x_0)$$ Then by continuity of $$\alpha$$ and compactness of $$\{0\} \times \overline{B} _{a/2}(x_0)$$, then there exists $$\epsilon>0$$ such that $$\alpha:(-\epsilon,\epsilon) \times B_{a/2}(x_0)\rightarrow B_a(x_0)$$ This last step I do not get, it does not seems trivial to me the existence of such $$\epsilon$$. My attempt of give a proof of such statement is the following:

First, since $$B_a(x_0)$$ is an open neighborhood of $$x_0$$ and $$\alpha$$ is continuous we have an open set $$V=(-\epsilon,\epsilon)\times B_\delta(x_0)$$ for some $$\epsilon,\delta>0$$, such that $$\alpha(V)\subset B_a(x_0)$$ Here compactness should play an important role to bound $$\delta$$, but I cannot see this. Can someone please give me a hint or full proof of this last statement?