Let $f\in\mathcal{C}^1(U,\Bbb{R}^n)$ and $x_0\in U$. Assume that there exists $c>0$ such that $[0,c)\subset J(x_0)$ where $J(x_0)$ is the maximal interval where there is a integral curve $\gamma_{x_0}$.
Then there exists an open neighborhood $V$ of $x_0$ and $k>0$ such that for all $x\in V$ the curve is defined on [0,c]
This is a theorem in my course (but I was not present today), the proof starts as follow:
By compactness of $[0,c]$ there exists $\varepsilon>0$ such that if $t\in[0,c]: \Vert x-\gamma_{x_0}(t)\Vert<\varepsilon$ then $x\in U$
I don't understand why the compactness gives us the existence of $\varepsilon>0.$