I am having trouble understanding the proof that asymptotic stability implies the existence of a strong Lyapunov function. Taken from the book "Differential Dynamical Systems", chapter 4, by James Meiss. The definitions are:
and the theorem is:
is a strong Lyapunov function. This is the start of the proof: 
I don't understand how asymptotic stability implies there exists such a "uniform" time $T(\rho)$. I do understand how for all $x\in U$ there exists some time $T(\rho, x)$ but don't understand how there exists a uniform $T(\rho)$ which does not depend on $x$.