So as per the title, I'm trying to prove that a function $f(n) = n^2 + 8n$ exists in $\Theta (n^2)$. What I'm having trouble with is the logic/concept behind doing so.
By definition, it would mean that:
$c_1n^2 \leq n^2 + 8n \leq c_2 n^2 $for constants $c1, c2$, and $N$ such for all $n\geq N$.
I get that. Now, do I just picking random constants that make the equality true? For example, the right side $O$ of the equality would be
$n^2 + 8n \leq c_2n^2$ for all $n \geq N$.
Now, if I let $N = 1$, I could let $n = 1$ also, and therefore the equality becomes $(1)^2 + 8(1) <= c_2(1)^2$, or $9 \leq c_2$. Therefore, $c_2$ could be $9$.
Now, I have proven that there is a $c_2$ and $N$ that makes the right side of the equality hold. I would then use the same $N$ for the left side of the equality, and choose $c_1 = 1$. Is that all is required to prove? How do I really know that my chosen N is correct, such that the equality holds for all $n \geq N$?
I guess what I am asking is if this is indeed a "proper" way to approach proving a function belongs to $\Theta$, or am I missing a concept? Thanks in advance.