Complete Vector field

Question

I am reading "Geometry of Differential Forms". We want to show that on a smooth compact manifold, vector fields are complete. We claim that there is an interval $(-\epsilon ~ ~\epsilon)$ of time where the integral curve exists on the whole manifold. Normally, the interval is a function of the point. I think the author is using something like the upper and lower limits are continuous functions of the point and hence have a maximum and a minimum. But, I cannot show this. The author appears to think this is obvious. I know there are other proofs using compactness but do you know anyway of showing this the way the author approaches it?

Answer 1

I found the the answer in the last page here:

http://www-personal.umich.edu/~wangzuoq/437W13/Notes/Lec%2012.pdf