# How to multiply using big $O$ notation

Suppose I have the following expression:

$$f(n) = \frac{\bigg(1+O(1/n)\bigg)}{\bigg(1+O(1/n)\bigg)\bigg(1+O(1/n)\bigg)}.$$

Is there a way to manipulate $$f$$ (i.e. multiply by terms involving $$n$$) in order to get an expression that looks like $$(1+O(1))$$?

If $$a_n,b_n,c_n\in O(1/n)$$, then \begin{align} \frac{1+a_n}{(1+b_n)(1+c_n)}-1 &=\frac{1+a_n-(1+b_n)(1+c_n)}{(1+b_n)(1+c_n)}\\ &=\frac{a_n-b_n-c_n-b_nc_n}{(1+b_n)(1+c_n)}\\ &\leq\frac{|a_n|+|b_n|+|c_n|+|b_n||c_n|}{|1+b_n||1+c_n|}\\ &=O(1/n) \end{align} hence $$f(n)=\frac{1+O(1/n)}{(1+O(1/n))(1+O(1/n))}=1+O(1/n)$$ consequently, $$f(n)=1+O(1)$$ as well.