Let
$x'(t)=f(t,x(t)), t\in(0,T)$ with $x(0)=x_0$
$f$ satifies the Lipschitz-condition $f(t,x)-f(t,y)\le L|x-y|$
$h\in (0,\frac{1}{L})$ is the step size and the approximation $x_k$ for $x(t_k)=hk$ is given by $x_k=x_{k-1}+hf(t_k,x_k)$.
Now I would be very interested how to derive the error
$|x_k-x(t_k)|\le\frac{1}{1-Lh}\left(|x_{k-1}-x(t_{k-1})|+\frac{h^2}{2} \max_{s\in [0,T]}|x''(s)|\right)$
I tried to look up it up in some numerical analysis books but it is always different