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 problem is simply a quadrature
because  $f(t − c/k, y(t − c/k))$ is known for $t \in (t_0 + c/k, t_0 + 2c/k]$.
$$t \in (t_0 + c/k, t_0 + 2c/k]\implies t-c/k\in\cdots$$
Same argument for the next intervals.
