I recently submitted this for homework. The question asked to give a big-O estimate for (1) below. I have included the feedback in bold. It seems the solution I proposed lacked a proof and I unsure what I need to do to prove it. I wasn't even aware I had to offer a proof for a question that asked to just give a big - O estimate. Do I need to show how I arrived at (2) by including a statement like if (n) = O(a(n)) and g(n) = O(b(n)) then f(n).g(n)=O(a(n).b(n))? Please could someone advise what a complete solution should look like as I would like to correct my error
$$(1) \space(n! + 2^{n+3})(111n^3 + 15\log(n^{201} +1))$$
$n! = O(n^{n})$ <<---"Better O(n!)"
$2^{n+3}=O(2^{n+3})$
$111n^{3}=O(n^{3})$
$15\log(n^{201} +1)= O(15\log n^{201})$
Therefore the dominant term appears to derive from $(n!)\cdot(111n^{3})$ which would give us the following:
(2) $\space O(n^{n+3}) \space or \space O(n!n^3)$ . <<-- "You need to prove it"
