If we know existence of solutions for $$ x'=f(t,x), \quad x(t_0)=x_0, $$ obtained by Peano's Theorem, and furthermore we know that the solution in unique ( we have not assumed satisfaction of Lipschitz condition by $f$).
How we can proof that the sequence in Peano's Theorem is converging uniformly ( in the proof of Peano's Theorem, by the help of Arzela-Ascoli theorem we only have at least a subsequence converging uniformly and not every subsequence).
I think that if I be able to show that the sequence is cauchy sequence and since we have a subsequence convergent, then every subsequence will be convergent. but I can't use the uniqueness to show chauchyness
The sequence of approximate solutions $v_n:I\to\mathbb R$ (or $\mathbb R^n$) constructed by Peano method possesses a uniformly convergent subsequence (due to Arzela-Ascoli as $\{v_n\}_{n\in\mathbb N}$ is uniformly bounded and uniformly equicontinuous), and the limit can only be the UNIQUE solution $v$ of the underlying IVP. This is also true for every subsequence of $\{v_n\}_{n\in\mathbb N}$ for the same reason.
Using the following fact:
If every subsequence of a sequence $\{f_n\}_{n\in\mathbb N}$ possesses a sub-subsequence converging to $f$, then $\{f_n\}_{n\in\mathbb N}$ also converges to $f$.
we obtain that the whole $\{v_n\}_{n\in\mathbb N}$ converges to $v$.