How to extract $O(h^2)$ from $f\left(t_{i+1},y_i+hf(t_i,y_i)+O(h^2)\right)$

Question

This is the formula of explicit Heun's method $$ y_{i+1}=y_i+hf\left(t_{i+1},y_i+hf(t_i,y_i)+O(h^2)\right)+O(h^3) $$ and I want to prove that this formula is $O(h^3)$.

Answer 1

$y_{i+1}=y_i+hf\left(t_{i+1},y_i+hf(t_i,y_i)+O(h^2)\right)+O(h^3)=y_i+h(f\left(t_{i+1},y_i+hf(t_i,y_i)\right)+f'(t_{i+1},y_i+hf(t_i,y_i))O(h^2))+O(h^3)=y_i+hf\left(t_{i+1},y_i+hf(t_i,y_i)\right)+O(h^3)$