Prove that the maximal interval of existence is finite This problem is making me crazy. I'm a math student and I'm stuck with this problem. Can anyone help me please? I don't even know how to start.
Given an Initial value problem
$ x'(t) = f(x(t))\\
x(0) = x_0 > 0 $
Where $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ is Lipschitz. Prove that the Maximal Interval of Existence is finite if and only if
$\int_{x_0}^{+ \infty} \frac{1}{f(s)} ds < \infty$

A hint: If $f$ were given explicitly you would solve this differential equation by the method of  "separation of variables". Look at the formulas  you get in this way.
