How to prove $3h_n^2+o(h_n^2)\in O(h_n^2)$

Question

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 $O(h_n^2)$?

Answer 1

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