Suppose we have $L-1$ time intervals between $t_0$ and $t_{L}=T,$ i.e., $t_0<\ldots<t_l<\ldots<t_{L}=T.$ We have an error $e_{t_l}$ describing the absolute error difference between a real function and an approximate function at $t_l,$ which is bounded by $e_{t_{l+1}}$ at $t_{l+1}$and other constant independent error terms $\epsilon_1,$ $\epsilon_2,$ $\epsilon_3$ and $\epsilon_4,$ which have no correlation with $e_{t_l}$ and $e_{t_{l+1}}$ and at any time $t_l$ such that
$$e_{t_l}\leq \epsilon_1 +\epsilon_2 +e_{t_{l+1}} + \epsilon_3$$ and $$e_{T}\leq \epsilon_1 +\epsilon_2 +\epsilon_4 + \epsilon_3.$$
If I want to describe $e_{t_0}$ in terms of $\epsilon_1,$ $\epsilon_2,$ $\epsilon_3$ and $\epsilon_4$ with a form of
$$e_{t_0}\leq (L-1)(\epsilon_1+\epsilon_2+\epsilon_3)+\epsilon_4,$$
I wondered if $e_{t_0}$ is bounded when $L\rightarrow \infty.$
If I replace $L$ by $\infty,$ $e_{t_0}$ is equal and less than $$(\infty)(\epsilon_1+\epsilon_2+\epsilon_3)+\epsilon_4,$$ $e_{t_0}$ is not bounded or is not defined as $\infty\cdot 0$ is not defined though.
Am I right about this?