I'm trying to understand the proof of picard-lindeloff theorem, it basically proves that if a function $f(t, y)$ is continuous in t and lipschitz continous in y, then there exists a unique solution of the initial value problem.
I know that I need to use successive approximation to prove the existence, and that's okay (I've just need to use picard iteration and function converges to a local solution in an interval), but then I don't understand why I need to use gronwall's lemma to prove the uniqueness, according to wikipedia
https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem
I need to take two solutions $\phi$ and $\psi$ and show that $|\phi(t) - \psi(t)| -> \phi(t) = \psi(t)$ and it proves global uniqueness, but I don't understand this step.