We propose a nice proof of Peano existence theorem. This theorem shows that the continuity of the vector field suffices for the existence of solutions to the ODE; ordinary nonlinear differential equations. We notice that this theorem does not guarantee the uniqueness of the solution.
Local solutions
Let $I$ be an interval of $\mathbb{R}$ and $\Omega$ an open set of $\mathbb{R}^d$ with $d\in\mathbb{N}^\ast$. Let $f:I\times \Omega\to \mathbb{R}^d$ be a continuous function. Let $(t_0,x_0)\in I\times\Omega$ and consider the Cauchy problem \begin{align*}(CP)\quad\begin{cases} \dot{u}(t)=f(t,u(t)),& t\in I,\cr u(t_0)=x_0.\end{cases}\end{align*}
A local solution to the Cauchy problem $(CP)$ is a couple $(J,\varphi)$, where $J$ is a subinterval of $I$ containing $t_0$, $\varphi:J\to \Omega$ is a function of class $C^1$ with $u(t_0)=x_0$ and satisfying the differential equation in $(CP)$.
The proof of Peano existence theorem
Peano’s theorem says that if $f$ is continuous then the Cauchy problem $(CP)$ admits a least a local solution $(J,u)$.
The idea of the proof of this theorem: Let $r>0$ such that the closed ball $\overline{B}(x_0,r)\subset \Omega$. By reducing $I$ to a compact, we can assume that $f$ is bounded by a constant $M>0$ on $I\times \overline{B}(x_0,r) $. We set \begin{align*}J:=I\cap \left[t_0-\frac{r}{M}, t_0+\frac{r}{M}\right].\end{align*}
Let $(f_k)_k$ be a sequence of functions of class $C^1$ from $\mathbb{R}\times \mathbb{R}^d$ to $\mathbb{R}^d$ be converging to $f$ on $J\times \overline{B}(x_0,r)$, bounded by $M$. Each $f_k$ is locally Lipschitz, we can then use a version of Cauchy-Lipschitz theorem which gives the existence of a solution $u_k:J\to \overline{B}(x_0,r)$ such that\begin{align*} \forall t\in J,\quad \dot{u}_k(t)=f_k(t,u_k(t)),\quad u_k(t_0)=x_0.\end{align*}
By taking the integral between $t_0$ and $t$, we obtain \begin{align*}\forall t\in J,\quad u_k(t)=x_0+\int^t_{t_0} f_k(s,u_k(s))ds.\end{align*}On the other hand, a simple argument shows that the sequence of function $(u_k)_k$ is equicontinuous and equi-bounded. Then by using Ascoli theorem, there exists a subsequence $(u_{n_k})_k$ of $(u_k)_k$ that converge on $J$ to an application $u$ continue on $J$ into $ \overline{B}(x_0,r) $. This is the local solution to the Cauchy problem $(CP)$.
The uniqueness of the solution is not assured by Peano’s theorem
The continuity of $f$ in the proof of the Peano theorem is not sufficient for the uniqueness of the solution.
The following example strengthens this remark: the Cauchy problem \begin{align*} x'(t)=\sqrt{x(t)},\quad x(0)=0.\end{align*}The problem admits two solution; the null solution and the following solution \begin{align*}t\mapsto \begin{cases}\frac{t^2}{4},& t\ge 0,\cr -\frac{t^2}{4},& t\le 0.\end{cases}\end{align*}
