Big O Notation and finding witnesses I am trying to figure out some stuff here with Big O Notation. I mean I understand the concept of it and can generally be able to tell what the efficiency of something is, but I do not really understand how to find witnesses. Here is an example of one that I need to do. Can anyone help me understand this? Thanks in advance! 
Example 1:
$n^2$ is $O(0.001n^3)$.
Example 2:
$25n^4 − 19n^3 + 13n^2 − 106n + 77$ is $O(n^4)$.

I'm not sure what it means to find witnesses, but nevertheless:


*

*$n^2 = o(n^3)$, as well as $O(n^3)$ since $\lim_{n \to \infty}\frac{n^2}{0.001 n^3}=0$.  


2.Do the same with the second expression: $\lim_{n \to \infty} \frac{a n^4 +b n^3 + c n+d}{s n^4} = \frac{a}{s}+o(1)=O(1)$ where $a,b,c,d,s$ are some constants.
