Proving that one function is big o of another? I'm working through a big-O problem and have the intuition to know the answer, but don't feel comfortable in my proof. I need to prove from definitions (i.e. proving that there exists two constants $c$ and $n_0$) that
$$3^{3\log(n) + \log \log(n)}\log(n^{450}) \in O(9^{2\log(n)}).$$
Now I end up selecting an $n_0$ value of 1, and a $c$ value of 1. The entire left side of my equation collapses into $0$. I'm left with $0 \leq 9^0 = 1$, which is true.
Now my questions: this seems far too easy. Am I really allowed to pick those constants myself? Also, the example I used just doesn't seem very rigorous, but it does make it clear how one side grows much faster than the other, does it not? Any advice would be greatly appreciated.
 A: You have checked the inequality only for $n_0$ you need to check it also for all $n>n_0$.
An strategy, among many, could be to divide both functions.
$$\begin{align}\frac{3^{3\log(n)+\log\log(n)}\log(n^{450})}{9^{
2log(n)}}&=\frac{3^{3\log(n)+\log\log(n)}\log(n^{450})}{3^{4
log(n)}}\\&=450\cdot3^{-\log(n)+\log\log(n)}\log(n)\\&=450\cdot3^{-\log(n)+\log\log(n)}3^{\frac{\log\log(n)}{\log(3)}}\\&=450\cdot3^{-\log(n)+\log\log(n)+\frac{\log\log(n)}{\log(3)}}\end{align}$$
And try to prove this is bounded for large $n$. So far this is the definition. What we can try to do now is to compute the limit as $n\to\infty$. If this limit is exists, as a finite limit, then it follows that the quotient is bounded.
In our particular case we can compute 
$$\begin{align}\log\log(n)-\log(n)+\frac{\log\log(n)}{\log(3)}&=\log\left(\frac{\log(n)}{n}\right)+\log(\log(n)^{1/\log(3)})\\&=\log\left(\frac{(\log(n))^{1+1/\log(3)}}{n}\right)\end{align}$$
Apply L'Hospital to the fraction inside the first $\log$. We get 
$$\left(1+\frac{1}{\log(3)}\right)\frac{(\log(n))^{1/\log(3)}}{n}$$
and again
$$\left(1+\frac{1}{\log(3)}\right)\frac{1}{\log(3)}\frac{1}{(\log(n))^{1-1/\log(3)}n}\to0$$
Therefore the original fraction tends to $450\cdot3^0=450$ and is therefore bounded. From this limit we know that any constant greater than $450$ should work for some large $n_0$.
