I am trying to recall a result regarding the approximation of ODEs.
Consider the ODE $x'=f(x)$ for some locally Lipshitz function $f$.
Consider the function $h_{n,t}(x) = x+f(x)*\frac{t}{n}$ and let $H_{n,t}(x) = h_{n,t}^n(x)$ where exponentiation by $n$ denotes composition of the function $n$ times. Let $G_t(x) = lim_{n\rightarrow\infty}H_{n,t}(x)$. I have the following questions:
1: Does $G_t(x)$ always exist? (eg: does the limit defining it exist)
2: Is $G_t(x)$ always a solution to the ODE?
3: What is the name of this result and where can I find a proof.
Also, I may have gotten the wrong condition on $f$, the condition might be differentiable, or maybe something else, I can't recall exactly.