Consider the ODE as follows: $$ y' = f(y,t),\;\; y(0) = y_0$$ and the numerical approximation is defined as follows: $$ y_{n+1} = y_{n} + \frac{h}{2}(f(y_n, t_n) + f(y_n + hf(y_n,t_n), t_{n+1}))$$ To proof $$ y(t_{n+1}) - y_{n+1} = C\cdot h^3$$ where of course $y(t_{n+1})$ is the analytic solution at $t_{n+1}$, I assumed $f(y,t) = ay$ with $a<0$ and it is A-stable, i.e. $y_n\xrightarrow{n\to\infty}0$ with some upper bound of $h$. I think we should use Taylor in two variables at some point but now sure how to proceed it.
I appreciate your reply in advance!
$f(y_n+h f(y_n,t_n),t_n+h)=f(y_n,t_n)+h f_y(y_n,t_n) f(y_n,t_n) + h f_t(y_n,t_n)+O(h^2)$. Plug in and simplify. Then compare with the Taylor series you expect.