Hadamard theorem proof problem at the first lemma in do Carmo's book

Question

I'm reading DoCarmo's book, Riemannian Geometry and i don't understand something. At page 149 first lemma

I don't understand why at the end he says that "It follows that for all $t>0$..."

Answer 1

If $\langle J,J\rangle (t)>\langle J,J \rangle(0)$ then by Lagrange mean value theorem there exist $y \in (0,t)$ such that: $$\langle J,J\rangle ' (y)=\frac{\langle J,J\rangle (t)-\langle J,J\rangle (0)}{t}>0$$ But then he proved that $\langle J,J\rangle ' (t_2)\ge\langle J,J\rangle ' (t_1)$ whenever $t_2>t_1$. This means that the derivative is non negative for all $t$ and the result follows.