Consider a real-valued sequence $\{h_n\}_n$ with $\lim_{n \rightarrow \infty}h_n=0$. Could you help me to simplify $(O(h_n^2))^2+3O(h_n^2)$?
My attempt:
(1) $(O(h_n^2))^2=h_n^4(O(1))^2=h_n^4(O(1))=O(h_n^4)$
(2) $3O(h_n^2)=O(h_n^2)$
(3) $(O(h_n^2))^2+3O(h_n^2)=O(h_n^4)+O(h_n^2)=?$
If $h$ is small, $O(h^4)+O(h^2) =O(h^2) $ since $\frac{h^4}{h^2} \to 0 $ as $h \to 0$.
If $h$ is large, $O(h^4)+O(h^2) =O(h^4) $ since $\frac{h^2}{h^4} \to 0 $ as $h \to \infty$.