Let $I_o=[t_0,t_0+T]\subset\mathbb R, T>0$, If $f\in C^0(I_0\times\mathbb R,\mathbb R)$ and satisfies the Lipschitz condition:
$\forall t\in I_0, \forall y,y^{*}\in\mathbb R:|f(t,y)-f(t,y^{*})|\le L|y-y^{*}|$ . Now assume that $\forall t\in I_0$ the following
integro-differential equation holds;
$\displaystyle y'(t)=f(t,y(t))+\int_{t_0}^{t}k(t,s)y(s)ds$, with $y_0$ given and $k\in C^0(I_0\times I_0,\mathbb R)$
estimate the following quantity $\displaystyle\epsilon_n=y(t_{n+1})-y(t_n)-h\Big(f(t_n,y(t_n))+h\sum_{i=0}^{n-1}k(t_n,t_i)y(t_i)\Big)$
from above such that; $\displaystyle\sum_{i=0}^{N-1}|\epsilon_i|\to0, N\to\infty$$\qquad$$(h=t_{i+1}-t_i,\forall i)$
My attempt is:
If I consider $\epsilon_n=y(t_{n+1})-y(t_n)- h\underbrace{\Big(f(t_n,y(t_n))+h\sum_{i=0}^{n-1}k(t_n,t_i)y(t_i)\Big)}_{\approx y'(t_n)}$
So I divide both sides by $h$ and get;
$\frac{\epsilon_n}{h}=\underbrace{\frac{y(t_n+h)-y(t_n)}{h}}_{\to y'(t_n)}-(y'(t_n))\to 0$
Hence $\epsilon_n=o(h)$
and since $\displaystyle N=\frac{T}{h}$
$\displaystyle\sum_{i=0}^{N-1}|\epsilon_i|\approx\frac{T}{h}o(h)=T\frac{o(h)}{h}\to 0$
but is it possible to majorize $\epsilon_n$ such that, I get a more concrete proof, without that approximations ?