I'm trying to get down how to prove that something is $O(\cdots)$ or $\Theta(\cdots)$ but no matter what I look at, I don't get the reasoning as to how I can come to an answer.
So here's a couple of examples I've looked at:
Show that for any real constants $a$ and $b$, where $b > 0$, $(n+a)^b = \Theta(n^b)$
Show that $\log_2(n!) = O(n\log _2 (n))$
For 1 I understand that I have to show:
$$c_1 n^b \leq (n+a)^b \leq c_2 n^b$$
Do I just plug in random numbers until something works? That's what all the sites I've looked at seem to be doing, but that doesn't make sense.
For 2 I know that I have to show:
$$0 \leq \log_2(n!) \leq c(n\log_2(n))$$
Again, where do I even start? I can't wrap my head around any of this. :( Thanks for your help and patience.