Understanding proof of Peano's existence theorem I'm studying the proof of Peano's existence theorem on this paper.
At page 5 it is said that the problem
$$\begin{cases} 
y(t) = y_0 & \forall t ∈ [t_0, t_0 + c/k] \\ 
y'(t) = f(t − c/k, y(t − c/k)) & \forall t ∈ (t_0 + c/k, t_0 + c]
\end{cases}
$$
has a unique solution. 
Can you explain me why? Thank you.

The proof proceeds stepwise. At the outset, the solution is given for $t\in[t_0,t_0+c/k]$. Note carefully that then the right hand side of the second equation $y'(t)=f(t-c/k,y(t-c/k))$ is known for $t\in[t_0+c/k,t_0+2c/k]$, so you can integrate:
$$y(t)=y(t_0+c/k)+\int_{t_0+c/k}^\tau f(\tau-c/k,y(\tau-c/k))\,d\tau,\qquad t\in[t_0+c/k,t_0+2c/k].$$
Now repeat the same idea, gaining a unique solution for $t$ in the next interval $[t_0+2c/k,t_0+3c/k]$, and so on.
