What is the result of $\frac{h^2}{2}O(h)+O(h^3)$

Question

Why and how in the following expression $$ y_{n+1}=y_n+hy^{\prime}_n+\frac{1}{2}\left[ \frac{y^{\prime}_{n+1}-y^{\prime}_n}{h}+O(h) \right]h^2+O(h^3) $$ $$\Rightarrow y_{n+1}=y_n+h\left( y^{\prime}_n+\frac{1}{2}y^{\prime}_{n+1}+\frac{1}{2}y^{\prime}_n \right)+O(h^3) $$ the $O(h)$ term eliminated?

Answer 1

Basically, it's because $h^2O(h)=O(h^3)$.