TITLE: How to prove $3h_n^2+o(h_n^2)\in O(h_n^2)$
QUESTION [1 upvotes]: Could you help to go through this operations with little-o and big-O notation? Consider the real-valued sequences $\{A_n\}_n$ and $\{h_n\}_n$ and suppose $\lim_{n \rightarrow \infty} h_n=0$. Let
$$
A_n=3h_n^2+o(h_n^2)
$$
Question: is $A_n \in O(h_n^2)$? 
My attempt: 
(1) $3h_n^2\in o(h_n)$
(2) Using (1), $A_n\in o(h_n)+o(h_n^2)=o(h_n)$
(3) $A_n\in o(h_n) \subset O(h_n) $
How can I get $O(h_n^2)$?

REPLY [3 votes]: Hint: $$\lim_{n\to\infty} \frac{3h_n^2+o(h_n^2)}{h_n^2}=\lim_{n\to\infty} \frac{3h_n^2}{h_n^2}+\lim_{n\to\infty} \frac{o(h_n^2)}{h_n^2}=3+\lim_{n\to\infty} o(1)=3$$
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