Lemme 2.4 in Morse theory by Milnor This is lemma 2.4 from "Morse theory" by Milnor ,with the prove 


I have some questions about this prove :
1) why  $\displaystyle\frac{dc}{dt}(f)=\lim_{h\rightarrow 0} \frac{fc(t+h)-fc(t)}{h}$ and who is $f$ ?
2)Why the differential equation has locally one solution which depends on the initial conditions ?
3) what is the purpose of the last step "let $\varepsilon _0>0,\dots$"
Please help me
Thank you .
 A: 1) Tangent vectors can be viewed as derivations on the space of smooth functions on a manifold.  The action of $\frac{dc}{dt}$ on a function is the direction derivative of the function in the direction $\frac{dc}{dt}$.  So $\frac{dc}{dt}(f) = \lim\limits_{h\to 0} \frac{f(c(t+h)) - f(c(t))}{h}$ by definition.
2)  This is a standard result in ODEs.  You could check the reference Milnor provides.  You can also find a proof in Lee's Introduction to Smooth Manifolds.  Here is a MathOverflow thread with some more references.
3) I am not sure what you mean by "the last step."  Do you mean the last two paragraphs?  Milnor has shown that "for each point in $M$ there exists a neighborhood $U$ and a number $\varepsilon>0$ so that" the differential equation generates a unique smooth flow ($1$-parameter group of diffeomorphisms) on $U$ for $|t|<\varepsilon$.    We want to show that each of these flows can be pieced together into a global flow on $M$.
The support of the vector field is compact, so we can cover it with finitely many neighborhoods which admit flows.  As the flow at each point is unique, we definitely get a global flow as long as $|t|$ is less than $\varepsilon$ defined above.  As we only need finitely many neighborhoods, there is a smallest such $\varepsilon$ called $\varepsilon_0$.  Now I know that for $|t| < \varepsilon_0$, there is a global flow.
