Please could someone review my proof of the following big-O estimate thanks
$(n^2+8)(n+1)$
f(n) is O(g(n)) if there are positive constants C and k such that:
(1)f(n) $\le Cg(n)$ whenever n>k
(2)f(n) is O(g(n)) $\equiv \space \exists C \exists k \forall n(n>k \rightarrow f(n) \le Cg(n))$
To prove big-O, we will choose values for C and k and prove n>k implies f(n) $\le$Cg(n)
First, show $f_1(n)=n^2+8 = O(n^2)$
Choose k=1. Assuming n>1, then
$ \frac {f_1(n)}{g_1(n)} =\frac {n^2 + 8}{n^2}< \frac{n^2 + 8n^2}{n^2}=9$
Choose c = 9. Note that $ 8< 8n^2$. Thus, $n^2+8$ is $O(n^2)$ because
$n^2 +8 \le 9n^2$ whenever n>1.
Second, show $f_2(n)=n + 1 = O(n)$
Choose k=1. Assuming n>1, then
$ \frac {f_2(n)}{g_2(n)} =\frac {n+1}{n}< \frac{n+n}{n}=2$
Choose c = 2. Note that $ 1< n$. Thus, $n+1$ is $O(n)$ because
$n+1 \le 2n$ whenever n>1
Using theorem: Suppose that $f_1(n)\space is\space O(g_1(n))\space and \space f_2(n) \space is \space O (g_2(n)). Then (f_1f_2)(n) \space is \space O(g_1(n)g_2(n)) $
$\therefore (n^2+8)(n+1) \space = O(n^3) $