I have seen disproofs of Big-O when $f(n)$ is a single term, such as $n^2$, but I am unsure of how to write a disproof when $f(n)$ has > 1 terms (ex. $n^7 - n^5$) and $g(n)$ also has multiple terms. I am really confused by our class notes, so I was wondering if someone can clarify. Thanks
Edit: I meant to say Disproof of a Big-O statement. I apologize for any confusion.
(1). If $f(n)\in O(g(n))$ and $g(n)\in O(h(n))$ then $f(n)\in O(h(n)).$
(2). If $f_1,.., f_k$ (with $k\in \Bbb N$) are each in $O(g),$ and $A_1,...,A_k$ are constants, then then $\sum_{i=1}^kA_if_i\in O(g).$
(3). A polynomial of degree $d$ is in $O(n^d).$ And is not in $O(n^e)$ for any $e<d$ unless the polynomial is the constant $0.$
For example $n^7+24n^5-n^4= n^7(1+\frac {24}{n^2}-\frac {1}{n^3}),$ which, for all sufficiently large $n,$ is between $n^7/2$ and $2n^7.$
Remark: (1) and (2) are proved directly from the definition of Big $O$. Definitions are the tools of this trade. When you don't see how to start, write the problem with terms like $O(g)$ written in full, as their def'ns , to see what exactly you are aiming for.
For example: "Is $n^7-n^5\in O(g)$?" becomes "Does there exist $M>0$ such that $|n^7-n^5|\le M|g(n)|$ for all sufficiently large $n$?"