I want to prove the following claim :
On a compact differentiable manifold $M$ , any vector field generates a $1$-parameter group of diffeomorphisms.
So we define the integral curve $(c^1,\dots,c^n):=c=c_p:I_p\to M$ ($I_p$ is an interval containing $0$) for some $p\in M$ and some vector field $\displaystyle X=X^i\frac{\partial}{\partial x^i}$ by $$\frac{dc^i}{dt}(t)=X^i(c(t))$$ in local coordinates. Then by Picard-Lindelöf, for some initial point $c_p(0)=p\in M$, one has a nbd $U$ of $p$ such that $$\begin{cases}\frac{dc_q}{dt}(t)=X(c_q(t))\\c_q(0)=q\end{cases}$$ for all $q\in U$. Define $\varphi_t(q):=c_q(t)$. Then defining the operation $$(\varphi_t\circ\varphi_s)(q)=\varphi_{t+s}(q)$$ for all $q\in U$ and $t,s,t+s\in I_q$, one defines a $1$-parameter group of diffeomorphisms.
My question is, why the compactness of $M$ is necessary in this case? And how we can use it to prove the claim? Any help is appreciated.